This document provides an overview of key probability concepts including:
(1) Definitions of random experiments, sample spaces, events, and probability;
(2) The addition and multiplication theorems and conditional probability;
(3) Mathematical expectation and probability distributions including the binomial, Poisson, and normal distributions. Examples are provided to illustrate key terminology and formulas.
This document discusses key concepts in probability. It defines basic terms like experiment, sample space, event, and probability. It provides examples of calculating probability for coin tosses and dice rolls using the classical method of dividing the number of ways an event can occur by the total number of possible outcomes. The document also discusses limitations of the classical method and introduces the empirical and subjective methods of determining probability based on observed frequencies and personal judgment respectively.
This document discusses probability and Bayes' theorem. It provides examples of basic probability concepts like the probability of a coin toss. It then defines conditional probability as the probability of an event given another event. Bayes' theorem is introduced as a way to revise a probability based on new information. An example problem demonstrates how to calculate the probability of rain given a weather forecast using Bayes' theorem.
Conditional probability is the probability of an event occurring given that another event has occurred. It is calculated as the probability of both events occurring divided by the probability of the first event. An example is given of calculating the probability of drawing two white balls in succession from an urn without replacement. The formula for conditional probability is derived as the probability of events A and B occurring divided by the probability of A. This is demonstrated using an example of finding the percentage of friends who like chocolate that also like strawberry.
This document provides an outline and slides for a presentation on statistical distributions. It begins with an introduction to frequency distributions, measures of central tendency, variability, z-scores, and theoretical distributions. Examples of different types of distributions are shown including normal, binomial, t and chi-square distributions. The document concludes with examples of how to apply these distributions to calculate probabilities and test hypotheses related to health data.
The document provides an overview of the binomial distribution including its basics, prerequisites, and examples. It defines a binomial experiment as having a fixed number of independent trials where each trial results in one of two possible outcomes (success or failure) with a constant probability. The document gives examples of flipping a coin and throwing a die to illustrate binomial experiments. It also provides notation used in binomial distributions and shows how to determine if an experiment follows a binomial distribution.
The document discusses the law of large numbers and central limit theorem. It states that as the sample size increases, the sample mean will get closer to the population mean. It also explains that the distribution of sample means will approach a normal distribution as the sample size increases, regardless of the population distribution. Additionally, it provides an example stating that if a large number of dice are rolled, the average value will be close to 3.5.
This document introduces some basic concepts of set theory, including:
1) Defining sets by listing elements or describing properties. Common sets include real numbers, integers, etc.
2) Basic set operations like union, intersection, difference, and complement.
3) Relationships between sets like subset, proper subset, and equality.
4) Other concepts like partitions, power sets, and Cartesian products involving ordered pairs from multiple sets.
This document provides an overview of key probability concepts including:
(1) Definitions of random experiments, sample spaces, events, and probability;
(2) The addition and multiplication theorems and conditional probability;
(3) Mathematical expectation and probability distributions including the binomial, Poisson, and normal distributions. Examples are provided to illustrate key terminology and formulas.
This document discusses key concepts in probability. It defines basic terms like experiment, sample space, event, and probability. It provides examples of calculating probability for coin tosses and dice rolls using the classical method of dividing the number of ways an event can occur by the total number of possible outcomes. The document also discusses limitations of the classical method and introduces the empirical and subjective methods of determining probability based on observed frequencies and personal judgment respectively.
This document discusses probability and Bayes' theorem. It provides examples of basic probability concepts like the probability of a coin toss. It then defines conditional probability as the probability of an event given another event. Bayes' theorem is introduced as a way to revise a probability based on new information. An example problem demonstrates how to calculate the probability of rain given a weather forecast using Bayes' theorem.
Conditional probability is the probability of an event occurring given that another event has occurred. It is calculated as the probability of both events occurring divided by the probability of the first event. An example is given of calculating the probability of drawing two white balls in succession from an urn without replacement. The formula for conditional probability is derived as the probability of events A and B occurring divided by the probability of A. This is demonstrated using an example of finding the percentage of friends who like chocolate that also like strawberry.
This document provides an outline and slides for a presentation on statistical distributions. It begins with an introduction to frequency distributions, measures of central tendency, variability, z-scores, and theoretical distributions. Examples of different types of distributions are shown including normal, binomial, t and chi-square distributions. The document concludes with examples of how to apply these distributions to calculate probabilities and test hypotheses related to health data.
The document provides an overview of the binomial distribution including its basics, prerequisites, and examples. It defines a binomial experiment as having a fixed number of independent trials where each trial results in one of two possible outcomes (success or failure) with a constant probability. The document gives examples of flipping a coin and throwing a die to illustrate binomial experiments. It also provides notation used in binomial distributions and shows how to determine if an experiment follows a binomial distribution.
The document discusses the law of large numbers and central limit theorem. It states that as the sample size increases, the sample mean will get closer to the population mean. It also explains that the distribution of sample means will approach a normal distribution as the sample size increases, regardless of the population distribution. Additionally, it provides an example stating that if a large number of dice are rolled, the average value will be close to 3.5.
This document introduces some basic concepts of set theory, including:
1) Defining sets by listing elements or describing properties. Common sets include real numbers, integers, etc.
2) Basic set operations like union, intersection, difference, and complement.
3) Relationships between sets like subset, proper subset, and equality.
4) Other concepts like partitions, power sets, and Cartesian products involving ordered pairs from multiple sets.
1) The document introduces basic concepts of probability such as sample spaces, events, outcomes, and how to calculate classical and empirical probabilities.
2) It discusses approaches to determining probability including classical, empirical, and subjective probabilities. Simulations can also be used to estimate probabilities.
3) Examples are provided to illustrate calculating probabilities using classical and empirical approaches for single and compound events with different sample spaces.
It is a consolidation of basic probability concepts worth understanding before attempting to apply probability concepts for predictions. The material is formed from different sources. ll the sources are acknowledged.
This document discusses types of probability and provides definitions and examples of key probability concepts. It begins with an introduction to probability theory and its applications. The document then defines terms like random experiments, sample spaces, events, favorable events, mutually exclusive events, and independent events. It describes three approaches to measuring probability: classical, frequency, and axiomatic. It concludes with theorems of probability and references.
Bayes' theorem describes the probability of an event based on prior knowledge of conditions related to the event. For example, a person's age can make the probability of them having cancer more accurate than without knowing their age. Bayesian inference applies Bayes' theorem to statistical analysis by updating probabilities based on new evidence. The example problem calculates probabilities of drawing a red ball from two bags with different numbers of red and black balls using Bayes' theorem. It finds the probability of a red ball being from bag A given that a red ball was drawn is 2/5 divided by the total probability of drawing a red ball from either bag.
This presentation provides an introduction to basic probability concepts. It defines probability as the study of randomness and uncertainty, and describes how probability was originally associated with games of chance. Key concepts discussed include random experiments, sample spaces, events, unions and intersections of events, and Venn diagrams. The presentation establishes the axioms of probability, including that a probability must be between 0 and 1, the probability of the sample space is 1, and probabilities of mutually exclusive events sum to the total probability. Formulas for computing probabilities of unions, intersections, and complements of events are also presented.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
The document provides an overview of key probability concepts including:
1. Random experiments, sample spaces, events, and the classification of events as simple, mutually exclusive, independent, and exhaustive.
2. The three main approaches to defining probability: classical, relative frequency, and subjective.
3. Important probability theorems like the addition rule, multiplication rule, and Bayes' theorem.
4. How to calculate probabilities of events using these theorems, including examples of finding probabilities of independent, dependent, mutually exclusive, and conditional events.
The addition theorem of probability states that if two events A and B are mutually exclusive, the probability of occurrence of either A or B is the sum of the individual probabilities of A and B, or P(A or B) = P(A) + P(B). The theorem can be extended to three or more mutually exclusive events. The document provides an example calculation for finding the probability that a randomly drawn ball will have a number that is a multiple of 5 or 9. It also provides an example where two people shooting at a target are not mutually exclusive events, so the formula is modified to P(A or B) = P(A) + P(B) - P(A and B).
This document defines key probability concepts and provides examples to illustrate them. It discusses that probability expresses the likelihood of events, and is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Examples are given to demonstrate calculating probabilities of independent and conditional events. Common terms are defined, such as experiment, outcome, sample space, and event. General probability rules and how probabilities are expressed are also covered.
The document discusses the axioms of probability and some basic properties. It defines three axioms for assigning probability values to events in a finite sample space: 1) a probability is between 0 and 1, 2) the probability of the entire sample space is 1, and 3) for mutually exclusive events, the total probability is the sum of the individual probabilities. It then modifies the third axiom for infinite sample spaces and lists five elementary properties that can be obtained from the axioms, such as the probability of the complement of an event being 1 minus the original probability. The document also gives an example problem calculating the probability of not getting a white ball from a bag.
The document discusses key concepts in probability, including:
1) Random phenomena involve outcomes that are unknown but have possible values. Trials produce outcomes that make up events within a sample space.
2) The Law of Large Numbers states that independent repeated events will have a relative frequency that approaches a single probability value.
3) Theoretical probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes, assuming all outcomes are equally likely.
This document discusses basic concepts of probability, including:
- The addition rule and multiplication rule for calculating probabilities of compound events.
- Events can be disjoint (mutually exclusive) or not disjoint.
- The probability of an event occurring or its complement must equal 1.
- How to calculate the probability of at least one occurrence of an event using the complement.
- When applying the multiplication rule, you must consider whether events are independent or dependent.
The document provides an overview of probability concepts including:
- Probability is a measure of how likely an event is, defined as the number of favorable outcomes divided by the total number of possible outcomes.
- Theoretical probability predicts outcomes without performing experiments, dealing with events as combinations of elementary outcomes.
- Random experiments may have different results each time while deterministic experiments always produce the same outcome.
- Elementary events are individual outcomes, and compound events combine multiple elementary outcomes.
- Theoretical probability of an event is the number of favorable elementary events divided by the total number of possible events.
- The probabilities of an event and its negation must sum to 1.
The document discusses probability theory and provides definitions and examples of key concepts like conditional probability and Bayes' theorem. It defines probability as the ratio of favorable events to total possible events. Conditional probability is the probability of an event given that another event has occurred. Bayes' theorem provides a way to update or revise beliefs based on new evidence and relates conditional probabilities. Examples are provided to illustrate concepts like conditional probability calculations.
The document provides an introduction to probability theory, including definitions of key terms like trial, event, exhaustive events, favorable events, independent events, mutually exclusive events, and equally likely events. It discusses three approaches to defining probability: classical, statistical, and axiomatic. The classical approach defines probability as the ratio of favorable cases to total possible cases. The statistical approach determines probabilities based on empirical observations over many trials. The axiomatic approach uses set theory and axioms to define probability without restrictions of previous approaches.
The document provides an introduction to probability. It discusses:
- What probability is and the definition of probability as a number between 0 and 1 that expresses the likelihood of an event occurring.
- A brief history of probability including its development in French society in the 1650s and key figures like James Bernoulli, Abraham De Moivre, and Pierre-Simon Laplace.
- Key terms used in probability like events, outcomes, sample space, theoretical probability, empirical probability, and subjective probability.
- The three types of probability: theoretical, empirical, and subjective probability.
- General probability rules including: the probability of impossible/certain events; the sum of all probabilities equaling 1; complements
The document discusses different types of relations between elements of sets. It defines relations as subsets of Cartesian products of sets and describes how relations can be represented using matrices or directed graphs. It then introduces various properties of relations such as reflexive, symmetric, transitive, and defines what it means for a relation to have each property. Composition of relations is also covered, along with how relation composition can be represented by matrix multiplication.
The document discusses smoking-related deaths in the United States each year, including 123,800 from lung cancer out of a total of 438,000 smoking-related deaths. It then defines conditional probability as the probability of one event occurring given that another event has already occurred. Using this definition, it calculates the conditional probability that a smoking-related death will be caused by lung cancer as 28%, based on the number of lung cancer deaths divided by the total number of smoking-related deaths.
Queuing theory is the mathematical study of waiting lines and delays. It examines properties like average wait time, number of servers, arrival and service rates. Queues form when demand for a service exceeds capacity. The simplest queuing system has two components - a queue and server - with attributes of inter-arrival and service times. Queuing models use Kendall notation to describe systems, and the M/M/1 model is commonly used to analyze average queue length, wait times, and probability of overflow for single server queues. Queuing theory has applications in fields like telecommunications, healthcare, and computer networking.
The document describes a queuing system of an online legal service that receives customer emails and has lawyers respond to them. Key details:
- Emails arrive at a rate of 10 per hour with a coefficient of variation of 1.
- One lawyer responds to emails, taking on average 5 minutes with a standard deviation of 4 minutes.
- The average customer wait time is calculated to be 20.5 minutes.
- With a 10 hour work day, a lawyer would receive about 100 emails.
- The lawyer would have 1.66 hours for other work when not responding to emails.
- Reducing the standard deviation of response times to 0.5 minutes would not change average wait or lawyer work time.
1) The document introduces basic concepts of probability such as sample spaces, events, outcomes, and how to calculate classical and empirical probabilities.
2) It discusses approaches to determining probability including classical, empirical, and subjective probabilities. Simulations can also be used to estimate probabilities.
3) Examples are provided to illustrate calculating probabilities using classical and empirical approaches for single and compound events with different sample spaces.
It is a consolidation of basic probability concepts worth understanding before attempting to apply probability concepts for predictions. The material is formed from different sources. ll the sources are acknowledged.
This document discusses types of probability and provides definitions and examples of key probability concepts. It begins with an introduction to probability theory and its applications. The document then defines terms like random experiments, sample spaces, events, favorable events, mutually exclusive events, and independent events. It describes three approaches to measuring probability: classical, frequency, and axiomatic. It concludes with theorems of probability and references.
Bayes' theorem describes the probability of an event based on prior knowledge of conditions related to the event. For example, a person's age can make the probability of them having cancer more accurate than without knowing their age. Bayesian inference applies Bayes' theorem to statistical analysis by updating probabilities based on new evidence. The example problem calculates probabilities of drawing a red ball from two bags with different numbers of red and black balls using Bayes' theorem. It finds the probability of a red ball being from bag A given that a red ball was drawn is 2/5 divided by the total probability of drawing a red ball from either bag.
This presentation provides an introduction to basic probability concepts. It defines probability as the study of randomness and uncertainty, and describes how probability was originally associated with games of chance. Key concepts discussed include random experiments, sample spaces, events, unions and intersections of events, and Venn diagrams. The presentation establishes the axioms of probability, including that a probability must be between 0 and 1, the probability of the sample space is 1, and probabilities of mutually exclusive events sum to the total probability. Formulas for computing probabilities of unions, intersections, and complements of events are also presented.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
The document provides an overview of key probability concepts including:
1. Random experiments, sample spaces, events, and the classification of events as simple, mutually exclusive, independent, and exhaustive.
2. The three main approaches to defining probability: classical, relative frequency, and subjective.
3. Important probability theorems like the addition rule, multiplication rule, and Bayes' theorem.
4. How to calculate probabilities of events using these theorems, including examples of finding probabilities of independent, dependent, mutually exclusive, and conditional events.
The addition theorem of probability states that if two events A and B are mutually exclusive, the probability of occurrence of either A or B is the sum of the individual probabilities of A and B, or P(A or B) = P(A) + P(B). The theorem can be extended to three or more mutually exclusive events. The document provides an example calculation for finding the probability that a randomly drawn ball will have a number that is a multiple of 5 or 9. It also provides an example where two people shooting at a target are not mutually exclusive events, so the formula is modified to P(A or B) = P(A) + P(B) - P(A and B).
This document defines key probability concepts and provides examples to illustrate them. It discusses that probability expresses the likelihood of events, and is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Examples are given to demonstrate calculating probabilities of independent and conditional events. Common terms are defined, such as experiment, outcome, sample space, and event. General probability rules and how probabilities are expressed are also covered.
The document discusses the axioms of probability and some basic properties. It defines three axioms for assigning probability values to events in a finite sample space: 1) a probability is between 0 and 1, 2) the probability of the entire sample space is 1, and 3) for mutually exclusive events, the total probability is the sum of the individual probabilities. It then modifies the third axiom for infinite sample spaces and lists five elementary properties that can be obtained from the axioms, such as the probability of the complement of an event being 1 minus the original probability. The document also gives an example problem calculating the probability of not getting a white ball from a bag.
The document discusses key concepts in probability, including:
1) Random phenomena involve outcomes that are unknown but have possible values. Trials produce outcomes that make up events within a sample space.
2) The Law of Large Numbers states that independent repeated events will have a relative frequency that approaches a single probability value.
3) Theoretical probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes, assuming all outcomes are equally likely.
This document discusses basic concepts of probability, including:
- The addition rule and multiplication rule for calculating probabilities of compound events.
- Events can be disjoint (mutually exclusive) or not disjoint.
- The probability of an event occurring or its complement must equal 1.
- How to calculate the probability of at least one occurrence of an event using the complement.
- When applying the multiplication rule, you must consider whether events are independent or dependent.
The document provides an overview of probability concepts including:
- Probability is a measure of how likely an event is, defined as the number of favorable outcomes divided by the total number of possible outcomes.
- Theoretical probability predicts outcomes without performing experiments, dealing with events as combinations of elementary outcomes.
- Random experiments may have different results each time while deterministic experiments always produce the same outcome.
- Elementary events are individual outcomes, and compound events combine multiple elementary outcomes.
- Theoretical probability of an event is the number of favorable elementary events divided by the total number of possible events.
- The probabilities of an event and its negation must sum to 1.
The document discusses probability theory and provides definitions and examples of key concepts like conditional probability and Bayes' theorem. It defines probability as the ratio of favorable events to total possible events. Conditional probability is the probability of an event given that another event has occurred. Bayes' theorem provides a way to update or revise beliefs based on new evidence and relates conditional probabilities. Examples are provided to illustrate concepts like conditional probability calculations.
The document provides an introduction to probability theory, including definitions of key terms like trial, event, exhaustive events, favorable events, independent events, mutually exclusive events, and equally likely events. It discusses three approaches to defining probability: classical, statistical, and axiomatic. The classical approach defines probability as the ratio of favorable cases to total possible cases. The statistical approach determines probabilities based on empirical observations over many trials. The axiomatic approach uses set theory and axioms to define probability without restrictions of previous approaches.
The document provides an introduction to probability. It discusses:
- What probability is and the definition of probability as a number between 0 and 1 that expresses the likelihood of an event occurring.
- A brief history of probability including its development in French society in the 1650s and key figures like James Bernoulli, Abraham De Moivre, and Pierre-Simon Laplace.
- Key terms used in probability like events, outcomes, sample space, theoretical probability, empirical probability, and subjective probability.
- The three types of probability: theoretical, empirical, and subjective probability.
- General probability rules including: the probability of impossible/certain events; the sum of all probabilities equaling 1; complements
The document discusses different types of relations between elements of sets. It defines relations as subsets of Cartesian products of sets and describes how relations can be represented using matrices or directed graphs. It then introduces various properties of relations such as reflexive, symmetric, transitive, and defines what it means for a relation to have each property. Composition of relations is also covered, along with how relation composition can be represented by matrix multiplication.
The document discusses smoking-related deaths in the United States each year, including 123,800 from lung cancer out of a total of 438,000 smoking-related deaths. It then defines conditional probability as the probability of one event occurring given that another event has already occurred. Using this definition, it calculates the conditional probability that a smoking-related death will be caused by lung cancer as 28%, based on the number of lung cancer deaths divided by the total number of smoking-related deaths.
Queuing theory is the mathematical study of waiting lines and delays. It examines properties like average wait time, number of servers, arrival and service rates. Queues form when demand for a service exceeds capacity. The simplest queuing system has two components - a queue and server - with attributes of inter-arrival and service times. Queuing models use Kendall notation to describe systems, and the M/M/1 model is commonly used to analyze average queue length, wait times, and probability of overflow for single server queues. Queuing theory has applications in fields like telecommunications, healthcare, and computer networking.
The document describes a queuing system of an online legal service that receives customer emails and has lawyers respond to them. Key details:
- Emails arrive at a rate of 10 per hour with a coefficient of variation of 1.
- One lawyer responds to emails, taking on average 5 minutes with a standard deviation of 4 minutes.
- The average customer wait time is calculated to be 20.5 minutes.
- With a 10 hour work day, a lawyer would receive about 100 emails.
- The lawyer would have 1.66 hours for other work when not responding to emails.
- Reducing the standard deviation of response times to 0.5 minutes would not change average wait or lawyer work time.
Queueing Theory is the mathematical study of waiting lines in systems where demand for service exceeds the available resources. A pioneer in the field was Agner Krarup Erlang who applied its principles to telecommunications. The document discusses key concepts in queueing theory including arrival and service processes, queue configurations, performance measures and examples of real-world applications. It also covers limitations of classical queueing models in fully representing complex real systems.
Queuing theory is used to model waiting lines in systems where demand fluctuates. It can be used to optimize resource allocation to minimize costs associated with customer wait times and unused service capacity. The key elements of a queuing system include arrivals, a queue or waiting line, service channels, and a service discipline for determining order of service. Customers arrive according to a Poisson distribution and service times follow an exponential distribution. The goal of queuing analysis is to determine the number of service channels needed to balance wait time costs and idle resource costs.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise boosts blood flow, releases endorphins, and promotes changes in the brain which help regulate emotions and stress levels.
Pushdown automata are a new form of finite automata that have increased language recognition capabilities through the addition of a pushdown stack. A pushdown automaton is defined by an input alphabet, input tape, stack alphabet, pushdown stack, start state, halt states, push states, and read states. It can perform push and pop operations on the stack to recognize nested or hierarchical language structures that regular finite automata cannot. The addition of a stack allows pushdown automata to recognize context-free languages while retaining the intuitive graphical representation of finite automata.
The document discusses various sorting algorithms:
1. Brute force algorithms like selection sort and bubble sort are described. Radix sort, which sorts elements based on digit positions, is also introduced.
2. Divide and conquer algorithms like merge sort and quicksort are mentioned. Merge sort works by dividing the list into halves and then merging the sorted halves.
3. The document concludes by stating divide and conquer is a common algorithm design strategy that breaks problems into subproblems.
Virtual Water Interactions in Transboundary Water Francesca Greco
This document discusses the importance of virtual water in transboundary cooperation and water politics. It analyzes two case studies: (1) export-led agricultural schemes in the Nile Basin that have shifted power balances, and (2) Jordan's Disi pipeline that was originally used for virtual water exports but now supplies domestic water. The author argues that virtual water flows, managed by private actors, can alter water allocation and international relations, making food-water interactions an important but often overlooked part of water policy analysis in transboundary basins. Accounting for virtual water trades and investments is key to understanding dynamics in major river systems worldwide.
Force water footprint & film screening g.k 1JALRAKSHAK
This document outlines activities for an event held by FORCE-HSBC on June 12, 2012 in Greater Kelash Part-1. It includes a water footprint pair up activity and a film screening activity to raise awareness about water conservation issues. The document ends by thanking participants.
This document provides an overview of water footprints and virtual water. It discusses that a water footprint accounts for both direct and indirect water usage, with indirect usage through consumption of goods and services making up 96% of our total water footprint. It defines key terms like virtual water, which is the water used to produce a product, and explains the three components that make up a water footprint - blue water from surface and groundwater, green water from rainfall, and grey water needed to dilute pollutants. The document aims to help readers understand their total water usage and impacts through considering both direct and virtual water in their consumption.
This document discusses changes in the balance of power in the Nile Basin, including the addition of South Sudan as a new riparian state, new infrastructure projects like the Grand Ethiopian Renaissance Dam, and Ethiopia taking a more assertive stance. It argues that upstream states are gaining bargaining power and challenging the historical status quo dominated by Egypt through counter-hegemonic actions. There is a debate around whether the Cooperative Framework Agreement and shift towards more bilateral and trilateral cooperation represents a new paradigm that could tip the balance of power towards upstream states.
The document discusses different types of scientific knowledge and approaches to water management. It describes modeling science as aiming to predict through modeling and make causal explanations to influence society. Interpretive science seeks subjective meanings and understanding through interpretation. Activist science aims to create social transformation through participatory research to address injustices. It notes that constructed knowledge will always overwhelm observed science, and politics is needed to deal with the ambiguity and uncertainty in managing complex social issues like water.
This document provides the goal, scope, and water accounting for a life cycle assessment of the author's personal water footprint over the course of one year. The goal is to determine the direct and indirect blue, green, and gray water consumption attributable to the author to assess sustainability. The scope outlines the functions, system boundaries, and allocation procedures. Water accounting is provided for the author's food, clothing, and household water use. The total water footprint is calculated to be over 1.6 million liters for food and nearly 300,000 liters for clothing per year.
Water Resource Reporting and Water Footprint from Marcellus Shale Development...Brian Rosa
This report analyzes water usage and waste disposal data from Marcellus Shale gas development in West Virginia and Pennsylvania between 2009-2012. It finds that over this period, nearly 9,000 horizontal gas wells were permitted using hydraulic fracturing, which led to increased gas production but also significant water usage and waste generation. The report calculates water footprints for gas production and identifies areas where state reporting requirements could be improved to better inform management of environmental impacts. It concludes that while regulation of the industry has strengthened water protections, continued improvements to data transparency are still needed.
This document discusses the concept of virtual water trade. It introduces how virtual water refers to the water used in the production of agricultural and industrial products. Countries can export virtual water through exporting water intensive goods, and this allows water-scarce nations to import virtual water. While real water trade between countries is difficult, virtual water trade through goods is more practical and can improve global water efficiency. The document outlines how virtual water content is calculated for different crop types and production systems.
Queueing theory and the M/M/1/∞/∞ model are discussed. An example queue has arrivals of 10, 25, 5, 15, 20 customers separated by service times of 35, 20, 60, 15, 134 units. The queue length over time is plotted, reaching a maximum of 5 customers. Waiting times between arrivals are given as random variables with values for 5 arrivals totaling 20 time units. The average queue length is calculated as the total time customers spend waiting divided by the time period.
This document provides an introduction to queueing theory. It discusses key concepts such as random variables, probability distributions, performance measures, Little's law and the PASTA property. It then examines several common queueing models including the M/M/1, M/M/c, M/Er/1, M/G/1 and G/M/1 queues. For each model it derives the equilibrium distribution and discusses measures like mean queue length and waiting time. The goal is to give an overview of basic queueing theory concepts and common single-server and multi-server queues.
Water Depletion/Affordability of Food - Presentation by Ashok Kumar Chapagain, Science Director, Water Footprint Network. This presentation was given as part of the 'Metrics of Sustainable Diets and Food Systems Workshop co-organized by Bioversity International and CIHEAM-IAMM, November 4th -5th 2014, Agropolis International, Montpellier
Visit 'Metrics of Sustainable Diets and Food Systems' Workshop webpage.
http://www.bioversityinternational.org/metrics-sustainable-diets-workshop/
This document provides an overview and introduction to a report on the water footprint of Italy. It discusses key concepts around virtual water and water footprinting. It notes that the water footprint of national production in Italy is around 70 billion m3 per year, with agriculture being the largest user at 85% of the total footprint. The focus of the report will be analyzing Italy's water use, promoting more sustainable management of water resources, and increasing awareness of virtual water flows and impacts on water systems.
Probability is a branch of mathematics that studies patterns of chance. It is used to quantify the likelihood of events occurring in experiments or other situations involving uncertainty. The probability of an event is expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty. Key concepts in probability include theoretical and experimental probability, sample spaces, events, mutually exclusive and exhaustive events, and rules like addition rules for calculating combined probabilities. Probability is applied in many fields including statistics, gambling, science, and machine learning.
Probability is a branch of mathematics used to quantify the likelihood of events occurring based on patterns of chance observed in experiments. It is defined as the ratio of favorable outcomes to total possible outcomes. Common probability terms include experiment, outcome, equally likely outcomes, sample space, event, and sample point. Probability calculations follow set rules and can be applied in various domains like gambling, science, and artificial intelligence.
This document discusses types of probability and human resource information systems (HRIS). It begins with an introduction to probability theory and its applications in statistics. It then defines key probability concepts like random experiments, sample spaces, events, outcomes, and approaches to measuring probability through classical, frequency, and axiomatic methods. The document also discusses uses of HRIS in functions like payroll, training, recruitment and its benefits in providing accurate data, analysis and decision making. However, it notes some HRIS challenges like costs and limited understanding. In conclusion, the document argues that while machines are created to serve humans, we are increasingly dependent on them for thought and action.
The document provides an introduction to probability concepts including sample spaces, events, mutually exclusive and exhaustive events, independent and dependent events, and formulas like the addition rule and multiplication rule. It explains terms used in probability like sample points, trials, outcomes, and experiments. Various approaches to probability are discussed including classical, statistical, subjective, and axiomatic approaches.
In this slide, variables types, probability theory behind the algorithms and its uses including distribution is explained. Also theorems like bayes theorem is also explained.
Probability is a mathematical measure of how likely events are to occur. It can be expressed as a fraction, decimal, or percentage between 0 and 1. A probability experiment involves possible outcomes that make up a sample space. An event is a subset of outcomes. Theoretical probability calculates the likelihood of an event based on equally likely outcomes. Experimental probability is based on observed frequencies. Subjective probability relies on estimates rather than calculations. As experiments are repeated, experimental probability approaches theoretical probability due to the law of large numbers.
This document provides an overview of key probability concepts including:
- Random experiments, outcomes, events, sample spaces, mutually exclusive and independent events
- Classical, relative frequency, subjective, and axiomatic approaches to defining probability
- Probability theorems including addition, multiplication, Bayes' theorem, and their applications to dependent and independent events
- Examples are provided to illustrate concepts like conditional probability and using Bayes' theorem to update probabilities with new information
This document provides an introduction to probability theory and different probability distributions. It begins with defining probability as a quantitative measure of the likelihood of events occurring. It then covers fundamental probability concepts like mutually exclusive events, additive and multiplicative laws of probability, and independent events. The document also introduces random variables and common probability distributions like the binomial, Poisson, and normal distributions. It provides examples of how each distribution is used and concludes with characteristics of the normal distribution.
STSTISTICS AND PROBABILITY THEORY .pptxVenuKumar65
The document discusses key concepts in probability theory including probability, random experiments, sample spaces, events, random variables, probability distributions, and Bayes' theorem. It covers the binomial, Poisson, and normal distributions and their characteristics and applications. Decision theory is introduced as analyzing choices under uncertainty involving defining problems, identifying outcomes, assessing criteria, and evaluating alternatives to make optimal decisions.
Probability refers to the likelihood of an event occurring, expressed as a value between 0 and 1. It is a branch of mathematics used to predict the chance of future events. There are different types of probability distributions that describe the chances of various outcomes in random experiments or events, such as the binomial, normal, and uniform distributions. Probability and statistics are related but distinct concepts, with probability focusing on chances and statistics handling the analysis of data.
The document defines key concepts in probability and hypothesis testing. It discusses probability as a numerical quantity between 0 and 1 that expresses the likelihood of an event. Different probability distributions are covered, including binomial, normal, and Poisson distributions. Hypothesis testing is defined as a methodology to either accept or reject a null hypothesis based on sample data. Types of hypotheses, terms used in testing like test statistics and p-values, and types of errors are also summarized.
This document introduces key concepts in probability:
- Probability is the likelihood of an event occurring, which can be measured numerically or described qualitatively.
- Events can be classified as exhaustive, favorable, mutually exclusive, equally likely, complementary, and independent.
- There are three approaches to defining probability: classical, frequency, and axiomatic. The classical approach defines probability as the number of favorable outcomes over the total number of possible outcomes. The frequency approach defines probability as the limit of the ratio of favorable outcomes to the total number of trials. The axiomatic approach defines probability based on axioms or statements assumed to be true.
- Key properties of probability include that the probability of an event is between 0
This document introduces key concepts in probability:
1. Probability is the likelihood of an event occurring, which can be expressed as a number or words like "impossible" or "likely".
2. Events can be classified as exhaustive, favorable, mutually exclusive, equally likely, complementary, and independent.
3. There are three approaches to defining probability: classical, frequency, and axiomatic. The classical approach defines probability as the number of favorable outcomes over the total number of possible outcomes. The frequency approach defines it as the limit of favorable outcomes over total trials. The axiomatic approach uses axioms like probabilities being between 0 and 1.
4. Several properties of probability are described, like the sum
CHAPTER 1 THEORY OF PROBABILITY AND STATISTICS.pptxanshujain54751
Probability theory is a branch of mathematics that uses concepts like sample space, probability distributions, and random variables to assign numerical likelihoods to the chances of outcomes occurring in random phenomena. It involves both theoretical and experimental approaches. Key aspects of probability theory include defining events and random variables, understanding independent and dependent events, and using formulas to calculate probabilities. Probability theory has various applications, like in finance to model markets, in product design to reduce failure probabilities, and in casinos to shape games of chance.
This document discusses probability theory and its applications. It begins by defining probability as a measure of how likely an event is to occur between 0 and 1. It then provides examples of calculating theoretical probability for simple events like a coin toss or dice roll. The document goes on to explain how probability theory is applied in many areas such as mathematics, statistics, science, and engineering. It provides examples of using probability for risk assessment in fields like finance, biology, and engineering reliability. Finally, it discusses how probability assessments influence decisions and have changed society.
Page 266LEARNING OBJECTIVES· Explain how researchers use inf.docxkarlhennesey
Page 266
LEARNING OBJECTIVES
· Explain how researchers use inferential statistics to evaluate sample data.
· Distinguish between the null hypothesis and the research hypothesis.
· Discuss probability in statistical inference, including the meaning of statistical significance.
· Describe the t test and explain the difference between one-tailed and two-tailed tests.
· Describe the F test, including systematic variance and error variance.
· Describe what a confidence interval tells you about your data.
· Distinguish between Type I and Type II errors.
· Discuss the factors that influence the probability of a Type II error.
· Discuss the reasons a researcher may obtain nonsignificant results.
· Define power of a statistical test.
· Describe the criteria for selecting an appropriate statistical test.
Page 267IN THE PREVIOUS CHAPTER, WE EXAMINED WAYS OF DESCRIBING THE RESULTS OF A STUDY USING DESCRIPTIVE STATISTICS AND A VARIETY OF GRAPHING TECHNIQUES. In addition to descriptive statistics, researchers use inferential statistics to draw more general conclusions about their data. In short, inferential statistics allow researchers to (a) assess just how confident they are that their results reflect what is true in the larger population and (b) assess the likelihood that their findings would still occur if their study was repeated over and over. In this chapter, we examine methods for doing so.
SAMPLES AND POPULATIONS
Inferential statistics are necessary because the results of a given study are based only on data obtained from a single sample of research participants. Researchers rarely, if ever, study entire populations; their findings are based on sample data. In addition to describing the sample data, we want to make statements about populations. Would the results hold up if the experiment were conducted repeatedly, each time with a new sample?
In the hypothetical experiment described in Chapter 12 (see Table 12.1), mean aggression scores were obtained in model and no-model conditions. These means are different: Children who observe an aggressive model subsequently behave more aggressively than children who do not see the model. Inferential statistics are used to determine whether the results match what would happen if we were to conduct the experiment again and again with multiple samples. In essence, we are asking whether we can infer that the difference in the sample means shown in Table 12.1 reflects a true difference in the population means.
Recall our discussion of this issue in Chapter 7 on the topic of survey data. A sample of people in your state might tell you that 57% prefer the Democratic candidate for an office and that 43% favor the Republican candidate. The report then says that these results are accurate to within 3 percentage points, with a 95% confidence level. This means that the researchers are very (95%) confident that, if they were able to study the entire population rather than a sample, the actual percentage who preferred th ...
It gives detail description about probability, types of probability, difference between mutually exclusive events and independent events, difference between conditional and unconditional probability and Bayes' theorem
This document provides an introduction to probability. It defines probability as a measure of how likely an event is to occur. Probability is expressed as a ratio of favorable outcomes to total possible outcomes. The key terms used in probability are defined, including event, outcome, sample space, and elementary events. The theoretical approach to probability is discussed, where probability is predicted without performing the experiment. Random experiments are described as those that may not produce the same outcome each time. Laws of probability are presented, such as a probability being between 0 and 1. Applications of probability in everyday life are mentioned, such as reliability testing of products. Two example probability problems are worked out.
Introduction to Statistics and ProbabilityBhavana Singh
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3. INTRODUCTION
Probability theory is a very fascinating subject which
can be studied at various mathematical levels.
Probability is the foundation of statistical theory and
applications.
To understand probability , it is best to envision an
experiment for which the outcome (result) is unknown.
Probability is the measure of how likely something will
occur.
It is the ratio of desired outcomes to total outcomes.
(# desired) / (# total)
4. TERMINOLOGIES
Random Experiment:
If an experiment or trial is repeated under
the same conditions for any number of times and it is
possible to count the total number of outcomes is called as
“Random Experiment”
Sample Space:
The set of all possible outcomes of a
random experiment is known as “Sample Space” and
denoted by set S. [this is similar to Universal set in Set
Theory] The outcomes of the random experiment are
called sample points or outcomes.
5. Random variable
Discrete Random Variable:
If the number of possible values of X
is finite or countably infinite then X is called a Discrete
Random Variable.
Continuous Random Variable:
A random variable X is called a
Continuous Random Variable if X takes all possible values
in an interval.
6. Events
Definition:
An ‘event’ is an outcome of a trial meeting
a specified set of conditions other words, event is a subset
of the sample space S.
Events are usually denoted by capital
letters
8. Exhaustive Events:
The total number of all possible elementary outcomes in a
random experiment is known as ‘exhaustive events’. In other
words, a set is said to be exhaustive, when no other possibilities
exists.
Favorable Events:
The elementary outcomes which entail or favor the happening
of an event is known as ‘favorable events’ i.e., the outcomes which
help in the occurrence of that event.
Mutually Exclusive Events:
Events are said to be ‘mutually exclusive’ if the occurrence of
an event totally prevents occurrence of all other events in a trial. In
other words, two events A and B cannot occur simultaneously.
9. Equally likely or Equi-probable Events:
Outcomes are said to be ‘equally likely’
if there is no reason to expect one outcome to occur in
preference to another. i.e., among all exhaustive outcomes,
each of them has equal chance of occurrence.
Complementary Events:
Let E denote occurrence of event. The
complement of E denotes the non occurrence of event E.
Complement of E is denoted by ‘Ē’.
Independent Events:
Two or more events are said to be
‘independent’, in a series of a trials if the outcome of one
event is does not affect the outcome of the other event or
vise versa.
10. Two or more events
If there are two or more events, you need
to consider if it is happening at the same time or one
after the other.
“And”
If the two events are happening at the same
time, you need to multiply the two probabilities together.
“Or”
If the two events are happening one after the
other, you need to add the two probabilities.
11. Probability distribution:
Binomial Distribution:
A random variable ‘x’ is said to follow
binomial distribution if it assumes only non negative
values and its probability mass function is given by
p(X=x) = 𝑛𝑐 𝑥 𝑝 𝑥
𝑞 𝑛−𝑥
12. Poisson Distribution:
A random variable ‘X’ taking non-
negative values is said to follow poisson distribution if its
probability mass function is given by
13. Geometric Distribution:
A random variable ‘x’ is said to
follow geometric distribution if it assumes non-
negative values and its probability mass function
is given by
P(X=x) = 𝑞 𝑥
p ,where x=0,1,2,3……
where
p+q=1 , then q=1-p;
0 ≤ p ≤ 1
14. Continuous Distribution:
Uniform Distribution:
A random variable ‘X’ is said to
follow uniform or rectangular distribution over an
interval(a,b) if its p.d.f is given by
15. Exponential Distribution:
A continuous Random Variable ‘X’ defined
in (0,∞) is said to follow an exponential
distribution with parameters λ if its p.d.f is given
by
f(x) = λ𝑒−λ𝑥
where λ > 0 and 0 ˂ x ˂ ∞
16. Gamma Distribution:
A continuous random variable ‘X’
is said to follow Gamma Distribution with parameters λ if
its p.d.f is given by
17. Example
If I flip a coin, what is the probability of getting heads?
What is the probability of getting tails?
Answer:
P(heads) = 1/2
P(tails) = 1/2
18. Another example
If I roll a number cube and flip a coin:
What is the probability I will get a heads and a 6?
What is the probability I will get a tails or a 3?
Answers
P(heads and 6) = 1/2 x 1/6 =1/12
P(tails or a 5) = 1/2 + 1/6 = 8/12 = 2/3
19. Practical applications
Probability in opinion poll:
The actual probability often applies to the
percentage of a large group. Suppose you know that 60
percent of the people in your community are Democrats,
30 percent are Republicans, and the remaining 10
percentage Independents or have another political
affiliation. If you randomly select one person from your
community, what’s the chance the person is a Democrat?
The chance is 60 percent. You can’t say that the person
is surely a Democrat because the chance is over 50
percent; the percentages just tell you that the person is
more likely to be a Democrat. Of course, after you ask
the person, he or she is either a Democrat or not; you
can’t be 60-percent Democrat.
20. Relative Frequency:
The approach is based on collecting data and, based on
that data, finding the percentage of time that an event
occurred. The percentage you find is the relative frequency
of that event — the number of times the event occurred
divided by the total number of observations made.
If you count 100 bird visits, and 27 of the visitors are
cardinals, you can say that for the period of time you
observe, 27 out of 100 visits or 27 percent, the relative
frequency — were made by cardinals. Now, if you have to
guess the probability that the next bird to visit is a
cardinal, 27 percent would be your best guess. You come
up with a probability based on relative frequency
21. Simulation:
Simulation approach is a process that creates data
by setting up a certain scenario, playing out that
scenario over and over many times, and looking at
the percentage of times a certain outcome occurs.
It’s different in three ways:
You create the data (usually with a computer); you
don’t collect it out in the real world.
The amount of data is typically much larger than
the amount you could observe in real life.
You use a certain model that scientists come up
with, and models have assumptions.
22. Statistics:
In statistics there is usually a collection of
random variables from which we make an observation and
then do something with the observation. The most common
situation is when the collection of random variables of
interest are mutually independent and with the same
distribution. Such a collection is called a random sample.
A statistic is a function of a random
sample that does not contain any unknown parameters.