This document provides an overview of continuous probability distributions covered in Lecture 5, including:
- Continuous random variables can take on uncountably infinite values within an interval, unlike discrete variables. Probability density functions (PDFs) are used instead of probabilities.
- The uniform, normal, and exponential distributions are introduced as examples of continuous distributions. Key properties like expected value and variance are discussed.
- The standard normal distribution is especially important, and its probabilities are provided in tables. Examples show how to calculate probabilities for normal distributions using the tables.
Statistik 1 6 distribusi probabilitas normalSelvin Hadi
This document discusses the key characteristics and concepts of the normal probability distribution. It outlines six goals related to understanding the normal distribution, its properties, calculating z-values, and using the normal distribution to approximate the binomial probability distribution. The key points covered include defining the mean, standard deviation, and shape of the normal curve; transforming variables to the standard normal distribution; and determining probabilities based on the areas under the normal curve.
The document discusses various methods for modeling input distributions in simulation models, including trace-driven simulation, empirical distributions, and fitting theoretical distributions to real data. It provides examples of several continuous and discrete probability distributions commonly used in simulation, including the exponential, normal, gamma, Weibull, binomial, and Poisson distributions. Key parameters and properties of each distribution are defined. Methods for selecting an appropriate input distribution based on summary statistics of real data are also presented.
This document summarizes solutions to odd-numbered homework problems from Chapter 4 of a statistics textbook. It covers topics like discrete vs. continuous random variables, probability distributions, the normal and binomial distributions, and how to calculate probabilities using the z-table. Examples include determining the type of random variable, finding probabilities of intervals for different distributions, and approximating binomial probabilities with the normal distribution for large n.
1. The document discusses the normal distribution and how to solve problems using normal distribution tables. It explains that the normal distribution is a theoretical probability curve where the area under the curve equals 1.
2. It provides key properties of the normal distribution including that 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
3. The document demonstrates how to use normal distribution tables to find probabilities and percentage points by standardizing data to the standard normal distribution with mean 0 and standard deviation 1.
This document outlines a lecture on continuous random variables and their probability distributions. It introduces probability density functions, cumulative distribution functions, and how to calculate the mean and variance of continuous random variables. It also covers specific continuous distributions like the uniform and normal distributions. Examples are provided to demonstrate calculating probabilities and standardizing normal random variables.
Discrete and continuous probability distributions ppt @ bec domsBabasab Patil
The document discusses various probability distributions including discrete and continuous distributions. It covers the binomial, hypergeometric, Poisson, and normal distributions. It provides the characteristics and formulas for each distribution and examples of how to calculate probabilities using the distributions.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
Statistik 1 6 distribusi probabilitas normalSelvin Hadi
This document discusses the key characteristics and concepts of the normal probability distribution. It outlines six goals related to understanding the normal distribution, its properties, calculating z-values, and using the normal distribution to approximate the binomial probability distribution. The key points covered include defining the mean, standard deviation, and shape of the normal curve; transforming variables to the standard normal distribution; and determining probabilities based on the areas under the normal curve.
The document discusses various methods for modeling input distributions in simulation models, including trace-driven simulation, empirical distributions, and fitting theoretical distributions to real data. It provides examples of several continuous and discrete probability distributions commonly used in simulation, including the exponential, normal, gamma, Weibull, binomial, and Poisson distributions. Key parameters and properties of each distribution are defined. Methods for selecting an appropriate input distribution based on summary statistics of real data are also presented.
This document summarizes solutions to odd-numbered homework problems from Chapter 4 of a statistics textbook. It covers topics like discrete vs. continuous random variables, probability distributions, the normal and binomial distributions, and how to calculate probabilities using the z-table. Examples include determining the type of random variable, finding probabilities of intervals for different distributions, and approximating binomial probabilities with the normal distribution for large n.
1. The document discusses the normal distribution and how to solve problems using normal distribution tables. It explains that the normal distribution is a theoretical probability curve where the area under the curve equals 1.
2. It provides key properties of the normal distribution including that 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
3. The document demonstrates how to use normal distribution tables to find probabilities and percentage points by standardizing data to the standard normal distribution with mean 0 and standard deviation 1.
This document outlines a lecture on continuous random variables and their probability distributions. It introduces probability density functions, cumulative distribution functions, and how to calculate the mean and variance of continuous random variables. It also covers specific continuous distributions like the uniform and normal distributions. Examples are provided to demonstrate calculating probabilities and standardizing normal random variables.
Discrete and continuous probability distributions ppt @ bec domsBabasab Patil
The document discusses various probability distributions including discrete and continuous distributions. It covers the binomial, hypergeometric, Poisson, and normal distributions. It provides the characteristics and formulas for each distribution and examples of how to calculate probabilities using the distributions.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
Binomial Distribution Part 5 deals with fitting & familiaring some concepts of B D under the complementary Statistics syllabus of University of Calicut in BSc core of Mathematics, Physics & Computer Science.
This document provides an overview of the key topics in Chapter 6 on the normal distribution, including:
1) It introduces continuous probability distributions and defines the normal distribution as the most important continuous probability distribution.
2) It explains how the normal distribution can be standardized to have a mean of 0 and standard deviation of 1, known as the standardized normal distribution.
3) It outlines the types of problems that will be solved using the normal distribution, including finding probabilities and percentiles for both the normal and standardized normal distribution.
8. normal distribution qt pgdm 1st semesterKaran Kukreja
The document discusses the normal distribution and its key properties. It explains that the normal distribution is a limiting case of the binomial distribution when the number of trials is large. It has a bell-shaped symmetrical curve centered around the mean. The normal distribution is uniquely defined by its mean and standard deviation. The document also covers how to convert between a normal distribution and the standard normal distribution and how to find probabilities using the standard normal distribution table.
4 2 continuous probability distributionnLama K Banna
Here are the steps to solve this problem:
a) Find the z-score corresponding to 115 mm Hg: (115 - 85)/13 = 2.31
The proportion that is NOT severely hypertensive is 1 - P(Z >= 2.31) = 1 - 0.0103 = 0.9897
b) Find the z-score corresponding to 90 mm Hg: (90 - 85)/13 = 0.3846
The proportion that will be asked to consult a physician is P(Z >= 0.3846) = 0.6507
c) Find the z-scores corresponding to the mildly hypertensive range:
(90 - 85)/13 = 0.3846
(
The document provides information about normal probability distributions and how to solve problems using normal distributions. It defines the normal distribution and standard normal distribution. It gives the equation for a normal distribution and how to standardize a normal variable. Examples are provided on finding probabilities and areas under the normal curve. The document also discusses using normal approximations to the binomial and Poisson distributions and provides continuity correction rules for such approximations.
The document discusses various techniques for identifying the appropriate probability distribution to model stochastic input data for simulation modeling. It covers histogram analysis to determine the shape of the data's distribution, quantile-quantile plots to evaluate how well a hypothesized distribution fits the data, parameter estimation methods for different distributions, and goodness-of-fit tests like the Kolmogorov-Smirnov test and chi-square test to statistically validate a distribution fit. An example involving vehicle arrival data illustrates these distribution-fitting procedures.
1. The document discusses the normal distribution and z-distribution (standard normal distribution). It provides definitions, properties, and examples of both.
2. The normal distribution is a bell-shaped curve that is symmetric around the mean. It is defined by its mean and standard deviation. The z-distribution is the standard normal distribution where the mean is 0 and standard deviation is 1.
3. Examples are provided to demonstrate how to calculate probabilities and find z-scores using the normal and z-distributions. Areas under the curve are calculated to find probabilities for various values in relation to the mean.
This document discusses probability distributions for random variables. It introduces discrete distributions like the binomial and Poisson distributions which are used for counting experiments. It also introduces continuous distributions like the normal distribution which are defined over continuous ranges of values. Key concepts covered include probability density functions, cumulative distribution functions, and how to relate random variables with specific parameters to standard distributions. Examples are provided to illustrate concepts like modeling the number of plant stems in a sampling area with a Poisson distribution.
discrete and continuous probability distributions pptbecdoms-120223034321-php...novrain1
This document discusses various probability distributions including discrete and continuous distributions. It covers the binomial, hypergeometric, Poisson, and normal distributions. For each distribution, it provides the characteristics and formulas, and examples of how to calculate probabilities using the distributions and probability tables or software. It also illustrates how the parameters impact the shape of the distributions. The goal is to help readers apply different probability distributions to problems and compute probabilities.
Big Data analysis involves building predictive models from high-dimensional data using techniques like variable selection, cross-validation, and regularization to avoid overfitting. The document discusses an example analyzing web browsing data to predict online spending, highlighting challenges with large numbers of variables. It also covers summarizing high-dimensional data through dimension reduction and model building for prediction versus causal inference.
The Binomial,poisson _ Normal Distribution (1).pptIjaz Manzoor
The document discusses key concepts related to probability distributions including the binomial, Poisson, and normal distributions. It provides overviews of each distribution including their parameters and properties. Examples are given to illustrate how to calculate probabilities using each distribution. Key figures are also included to visualize the different distributions.
Python is an interpreted, object-oriented, high-level programming language with dynamic semantics. Its high-level built in data structures, combined with dynamic typing and dynamic binding, make it very attractive for Rapid Application Development, as well as for use as a scripting or glue language to connect existing components together
This document discusses the normal distribution and related concepts. It begins with an introduction to the normal distribution and its properties. It then covers the probability density function and cumulative distribution function of the normal distribution. The rest of the document discusses key properties like the 68-95-99.7 rule, using the standard normal distribution, and how to determine if a data set follows a normal distribution including using a normal probability plot. Examples are provided throughout to illustrate the concepts.
When modeling a system, encountering missing data is common.
What shall a modeler do in the case of unknown or missing information?
When dealing with missing data, it is critical to make correct assumptions to ensure that the system is accurate.
One must common strategy for handling such situations is calculate the average of available data for the similar existing systems (i.e., creating sampling data).
Use this average as a reasonable estimate for the missing value.
This document provides an overview of key concepts in probability and probability distributions. It introduces random variables and their probability distributions, and covers discrete and continuous random variables. Specific probability distributions discussed include the binomial, Poisson, and normal distributions. Expected value and variance are defined as measures of the central tendency and variability of random variables. Examples are provided to illustrate calculating probabilities and parameters for different probability distributions.
This document provides an overview of key concepts in probability and statistics including:
1) Definitions of random variables, discrete and continuous distributions. Discrete variables can take countable values while continuous can take any value in an interval.
2) Common probability distributions like the binomial, Poisson, uniform, and normal distributions. Formulas are provided for the probability mass/density functions and calculating mean, variance, and probability.
3) The exponential distribution with applications like waiting times. Its probability density function and formulas for mean and variance are defined.
Probability
Random variables and Probability Distributions
The Normal Probability Distributions and Related Distributions
Sampling Distributions for Samples from a Normal Population
Classical Statistical Inferences
Properties of Estimators
Testing of Hypotheses
Relationship between Confidence Interval Procedures and Tests of Hypotheses.
The document discusses binary dependent variable regression models, including the linear probability model, probit model, and logit model. It uses a dataset on mortgage applications to demonstrate these models. Key points:
- The linear probability model predicts probabilities outside the valid 0-1 range, while probit and logit ensure predicted probabilities fall between 0-1.
- Probit regression models the probability of an event using the cumulative standard normal distribution. Logit uses the logistic distribution.
- Estimating these models on mortgage application data, the coefficients represent the effect of debt-to-income ratio and applicant race on the probability of denial.
- Predicted probabilities can be calculated and compared to assess differences across groups, like
This document provides an overview of the Poisson distribution and other special probability distributions. It discusses:
1) The Poisson distribution and its properties, including how it can model rare, independent events over time periods. Examples of how to calculate probabilities using the Poisson are provided.
2) Other discrete distributions like the binomial, negative binomial, and hypergeometric.
3) Continuous distributions like the uniform, exponential, gamma, chi-square, and Weibull distributions. Applications and properties of each are summarized.
4) Comparisons between distributions like how the Poisson approximates the binomial for large values. Overall, the document introduces several important probability distributions used in statistics.
2 Review of Statistics. 2 Review of Statistics.WeihanKhor2
This document provides an overview of discrete probability distributions, including the binomial and Poisson distributions.
1) It defines key concepts such as random variables, probability mass functions, and expected value as they relate to discrete random variables. 2) The binomial distribution describes independent Bernoulli trials with a constant probability of success, and is used to calculate probabilities of outcomes from events like coin flips. 3) The Poisson distribution approximates the binomial when the number of trials is large and the probability of success is small. It models rare, independent events with a constant average rate and can be used for problems involving traffic accidents or natural disasters.
A375 Example Taste the taste of the Lord, the taste of the Lord The taste of...franktsao4
It seems that current missionary work requires spending a lot of money, preparing a lot of materials, and traveling to far away places, so that it feels like missionary work. But what was the result they brought back? It's just a lot of photos of activities, fun eating, drinking and some playing games. And then we have to do the same thing next year, never ending. The church once mentioned that a certain missionary would go to the field where she used to work before the end of his life. It seemed that if she had not gone, no one would be willing to go. The reason why these missionary work is so difficult is that no one obeys God’s words, and the Bible is not the main content during missionary work, because in the eyes of those who do not obey God’s words, the Bible is just words and cannot be connected with life, so Reading out God's words is boring because it doesn't have any life experience, so it cannot be connected with human life. I will give a few examples in the hope that this situation can be changed. A375
The Book of Samuel is a book in the Hebrew Bible, found as two books in the Old Testament. The book is part of the Deuteronomistic history, a series of books that constitute a theological history of the Israelites and that aim to explain God's law for Israel under the guidance of the prophets.
Binomial Distribution Part 5 deals with fitting & familiaring some concepts of B D under the complementary Statistics syllabus of University of Calicut in BSc core of Mathematics, Physics & Computer Science.
This document provides an overview of the key topics in Chapter 6 on the normal distribution, including:
1) It introduces continuous probability distributions and defines the normal distribution as the most important continuous probability distribution.
2) It explains how the normal distribution can be standardized to have a mean of 0 and standard deviation of 1, known as the standardized normal distribution.
3) It outlines the types of problems that will be solved using the normal distribution, including finding probabilities and percentiles for both the normal and standardized normal distribution.
8. normal distribution qt pgdm 1st semesterKaran Kukreja
The document discusses the normal distribution and its key properties. It explains that the normal distribution is a limiting case of the binomial distribution when the number of trials is large. It has a bell-shaped symmetrical curve centered around the mean. The normal distribution is uniquely defined by its mean and standard deviation. The document also covers how to convert between a normal distribution and the standard normal distribution and how to find probabilities using the standard normal distribution table.
4 2 continuous probability distributionnLama K Banna
Here are the steps to solve this problem:
a) Find the z-score corresponding to 115 mm Hg: (115 - 85)/13 = 2.31
The proportion that is NOT severely hypertensive is 1 - P(Z >= 2.31) = 1 - 0.0103 = 0.9897
b) Find the z-score corresponding to 90 mm Hg: (90 - 85)/13 = 0.3846
The proportion that will be asked to consult a physician is P(Z >= 0.3846) = 0.6507
c) Find the z-scores corresponding to the mildly hypertensive range:
(90 - 85)/13 = 0.3846
(
The document provides information about normal probability distributions and how to solve problems using normal distributions. It defines the normal distribution and standard normal distribution. It gives the equation for a normal distribution and how to standardize a normal variable. Examples are provided on finding probabilities and areas under the normal curve. The document also discusses using normal approximations to the binomial and Poisson distributions and provides continuity correction rules for such approximations.
The document discusses various techniques for identifying the appropriate probability distribution to model stochastic input data for simulation modeling. It covers histogram analysis to determine the shape of the data's distribution, quantile-quantile plots to evaluate how well a hypothesized distribution fits the data, parameter estimation methods for different distributions, and goodness-of-fit tests like the Kolmogorov-Smirnov test and chi-square test to statistically validate a distribution fit. An example involving vehicle arrival data illustrates these distribution-fitting procedures.
1. The document discusses the normal distribution and z-distribution (standard normal distribution). It provides definitions, properties, and examples of both.
2. The normal distribution is a bell-shaped curve that is symmetric around the mean. It is defined by its mean and standard deviation. The z-distribution is the standard normal distribution where the mean is 0 and standard deviation is 1.
3. Examples are provided to demonstrate how to calculate probabilities and find z-scores using the normal and z-distributions. Areas under the curve are calculated to find probabilities for various values in relation to the mean.
This document discusses probability distributions for random variables. It introduces discrete distributions like the binomial and Poisson distributions which are used for counting experiments. It also introduces continuous distributions like the normal distribution which are defined over continuous ranges of values. Key concepts covered include probability density functions, cumulative distribution functions, and how to relate random variables with specific parameters to standard distributions. Examples are provided to illustrate concepts like modeling the number of plant stems in a sampling area with a Poisson distribution.
discrete and continuous probability distributions pptbecdoms-120223034321-php...novrain1
This document discusses various probability distributions including discrete and continuous distributions. It covers the binomial, hypergeometric, Poisson, and normal distributions. For each distribution, it provides the characteristics and formulas, and examples of how to calculate probabilities using the distributions and probability tables or software. It also illustrates how the parameters impact the shape of the distributions. The goal is to help readers apply different probability distributions to problems and compute probabilities.
Big Data analysis involves building predictive models from high-dimensional data using techniques like variable selection, cross-validation, and regularization to avoid overfitting. The document discusses an example analyzing web browsing data to predict online spending, highlighting challenges with large numbers of variables. It also covers summarizing high-dimensional data through dimension reduction and model building for prediction versus causal inference.
The Binomial,poisson _ Normal Distribution (1).pptIjaz Manzoor
The document discusses key concepts related to probability distributions including the binomial, Poisson, and normal distributions. It provides overviews of each distribution including their parameters and properties. Examples are given to illustrate how to calculate probabilities using each distribution. Key figures are also included to visualize the different distributions.
Python is an interpreted, object-oriented, high-level programming language with dynamic semantics. Its high-level built in data structures, combined with dynamic typing and dynamic binding, make it very attractive for Rapid Application Development, as well as for use as a scripting or glue language to connect existing components together
This document discusses the normal distribution and related concepts. It begins with an introduction to the normal distribution and its properties. It then covers the probability density function and cumulative distribution function of the normal distribution. The rest of the document discusses key properties like the 68-95-99.7 rule, using the standard normal distribution, and how to determine if a data set follows a normal distribution including using a normal probability plot. Examples are provided throughout to illustrate the concepts.
When modeling a system, encountering missing data is common.
What shall a modeler do in the case of unknown or missing information?
When dealing with missing data, it is critical to make correct assumptions to ensure that the system is accurate.
One must common strategy for handling such situations is calculate the average of available data for the similar existing systems (i.e., creating sampling data).
Use this average as a reasonable estimate for the missing value.
This document provides an overview of key concepts in probability and probability distributions. It introduces random variables and their probability distributions, and covers discrete and continuous random variables. Specific probability distributions discussed include the binomial, Poisson, and normal distributions. Expected value and variance are defined as measures of the central tendency and variability of random variables. Examples are provided to illustrate calculating probabilities and parameters for different probability distributions.
This document provides an overview of key concepts in probability and statistics including:
1) Definitions of random variables, discrete and continuous distributions. Discrete variables can take countable values while continuous can take any value in an interval.
2) Common probability distributions like the binomial, Poisson, uniform, and normal distributions. Formulas are provided for the probability mass/density functions and calculating mean, variance, and probability.
3) The exponential distribution with applications like waiting times. Its probability density function and formulas for mean and variance are defined.
Probability
Random variables and Probability Distributions
The Normal Probability Distributions and Related Distributions
Sampling Distributions for Samples from a Normal Population
Classical Statistical Inferences
Properties of Estimators
Testing of Hypotheses
Relationship between Confidence Interval Procedures and Tests of Hypotheses.
The document discusses binary dependent variable regression models, including the linear probability model, probit model, and logit model. It uses a dataset on mortgage applications to demonstrate these models. Key points:
- The linear probability model predicts probabilities outside the valid 0-1 range, while probit and logit ensure predicted probabilities fall between 0-1.
- Probit regression models the probability of an event using the cumulative standard normal distribution. Logit uses the logistic distribution.
- Estimating these models on mortgage application data, the coefficients represent the effect of debt-to-income ratio and applicant race on the probability of denial.
- Predicted probabilities can be calculated and compared to assess differences across groups, like
This document provides an overview of the Poisson distribution and other special probability distributions. It discusses:
1) The Poisson distribution and its properties, including how it can model rare, independent events over time periods. Examples of how to calculate probabilities using the Poisson are provided.
2) Other discrete distributions like the binomial, negative binomial, and hypergeometric.
3) Continuous distributions like the uniform, exponential, gamma, chi-square, and Weibull distributions. Applications and properties of each are summarized.
4) Comparisons between distributions like how the Poisson approximates the binomial for large values. Overall, the document introduces several important probability distributions used in statistics.
2 Review of Statistics. 2 Review of Statistics.WeihanKhor2
This document provides an overview of discrete probability distributions, including the binomial and Poisson distributions.
1) It defines key concepts such as random variables, probability mass functions, and expected value as they relate to discrete random variables. 2) The binomial distribution describes independent Bernoulli trials with a constant probability of success, and is used to calculate probabilities of outcomes from events like coin flips. 3) The Poisson distribution approximates the binomial when the number of trials is large and the probability of success is small. It models rare, independent events with a constant average rate and can be used for problems involving traffic accidents or natural disasters.
A375 Example Taste the taste of the Lord, the taste of the Lord The taste of...franktsao4
It seems that current missionary work requires spending a lot of money, preparing a lot of materials, and traveling to far away places, so that it feels like missionary work. But what was the result they brought back? It's just a lot of photos of activities, fun eating, drinking and some playing games. And then we have to do the same thing next year, never ending. The church once mentioned that a certain missionary would go to the field where she used to work before the end of his life. It seemed that if she had not gone, no one would be willing to go. The reason why these missionary work is so difficult is that no one obeys God’s words, and the Bible is not the main content during missionary work, because in the eyes of those who do not obey God’s words, the Bible is just words and cannot be connected with life, so Reading out God's words is boring because it doesn't have any life experience, so it cannot be connected with human life. I will give a few examples in the hope that this situation can be changed. A375
The Book of Samuel is a book in the Hebrew Bible, found as two books in the Old Testament. The book is part of the Deuteronomistic history, a series of books that constitute a theological history of the Israelites and that aim to explain God's law for Israel under the guidance of the prophets.
The forces involved in this witchcraft spell will re-establish the loving bond between you and help to build a strong, loving relationship from which to start anew. Despite any previous hardships or problems, the spell work will re-establish the strong bonds of friendship and love upon which the marriage and relationship originated. Have faith, these stop divorce and stop separation spells are extremely powerful and will reconnect you and your partner in a strong and harmonious relationship.
My ritual will not only stop separation and divorce, but rebuild a strong bond between you and your partner that is based on truth, honesty, and unconditional love. For an even stronger effect, you may want to consider using the Eternal Love Bond spell to ensure your relationship and love will last through all tests of time. If you have not yet determined if your partner is considering separation or divorce, but are aware of rifts in the relationship, try the Love Spells to remove problems in a relationship or marriage. Keep in mind that all my love spells are 100% customized and that you'll only need 1 spell to address all problems/wishes.
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A Free eBook ~ Valuable LIFE Lessons to Learn ( 5 Sets of Presentations)...OH TEIK BIN
A free eBook comprising 5 sets of PowerPoint presentations of meaningful stories /Inspirational pieces that teach important Dhamma/Life lessons. For reflection and practice to develop the mind to grow in love, compassion and wisdom. The texts are in English and Chinese.
My other free eBooks can be obtained from the following Links:
https://www.slideshare.net/ohteikbin/presentations
https://www.slideshare.net/ohteikbin/documents
Heartfulness Magazine - June 2024 (Volume 9, Issue 6)heartfulness
Dear readers,
This month we continue with more inspiring talks from the Global Spirituality Mahotsav that was held from March 14 to 17, 2024, at Kanha Shanti Vanam.
We hear from Daaji on lifestyle and yoga in honor of International Day of Yoga, June 21, 2024. We also hear from Professor Bhavani Rao, Dean at Amrita Vishwa Vidyapeetham University, on spirituality in action, the Venerable BhikkuSanghasena on how to be an ambassador for compassion, Dr. Tony Nader on the Maharishi Effect, Swami Mukundananda on the crossroads of modernization, Tejinder Kaur Basra on the purpose of work, the Venerable GesheDorjiDamdul on the psychology of peace, the Rt. Hon. Patricia Scotland, KC, Secretary-General of the Commonwealth, on how we are all related, and world-renowned violinist KumareshRajagopalan on the uplifting mysteries of music.
Dr. Prasad Veluthanar shares an Ayurvedic perspective on treating autism, Dr. IchakAdizes helps us navigate disagreements at work, Sravan Banda celebrates World Environment Day by sharing some tips on land restoration, and Sara Bubber tells our children another inspiring story and challenges them with some fun facts and riddles.
Happy reading,
The editors
Sanatan Vastu | Experience Great Living | Vastu ExpertSanatan Vastu
Santan Vastu Provides Vedic astrology courses & Vastu remedies, If you are searching Vastu for home, Vastu for kitchen, Vastu for house, Vastu for Office & Factory. Best Vastu in Bahadurgarh. Best Vastu in Delhi NCR
The Enchantment and Shadows_ Unveiling the Mysteries of Magic and Black Magic...Phoenix O
This manual will guide you through basic skills and tasks to help you get started with various aspects of Magic. Each section is designed to be easy to follow, with step-by-step instructions.
The Book of Ruth is included in the third division, or the Writings, of the Hebrew Bible. In most Christian canons it is treated as one of the historical books and placed between Judges and 1 Samuel.
Trusting God's Providence | Verse: Romans 8: 28-31JL de Belen
Trusting God's Providence.
Providence - God’s active preservation and care over His creation. God is both the Creator and the Sustainer of all things Heb. 1:2-3; Col. 1:17
-God keep His promises.
-God’s general providence is toward all creation
- All things were made through Him
God’s special providence is toward His children.
We may suffer now, but joy can and will come
God can see what we cannot see
Protector & Destroyer: Agni Dev (The Hindu God of Fire)Exotic India
So let us turn the pages of ancient Indian literature and get to know more about Agni, the mighty purifier of all things, worshipped in Indian culture as a God since the Vedic time.
The Hope of Salvation - Jude 1:24-25 - MessageCole Hartman
Jude gives us hope at the end of a dark letter. In a dark world like today, we need the light of Christ to shine brighter and brighter. Jude shows us where to fix our focus so we can be filled with God's goodness and glory. Join us to explore this incredible passage.
2. Lect05 Continuous Probability Distributions 2
Agenda
◼ Lecture 4: discrete probability distr. (Keller Ch.7)
◼ random variables and probability distributions
◼ expected value and variance (discrete random variable)
◼ binomial distribution and Poisson distribution
◼ Lecture 5: continuous probability distr. (Keller Ch.8)
◼ cont. prob. distr. and probability density functions
◼ uniform distribution (continuous)
◼ normal distribution and use of standard normal table
◼ exponential distribution and its relation to Poisson distr.
N.B. Sect.8.4: self-reading (student t, chi-quared and F
distribution); will be discussed in later sessions
3. Lect05 Continuous Probability Distributions 3
Continuous random variables
– Sect.8.1
◼ Discrete random variables:
◼ finite number of possible values (e.g. binomial variable; x =
0, 1, 2, …, n)
◼ countably infinite number of possible values (e.g. Poisson
variable; x = 0, 1, ...)
◼ Continous random variables:
◼ uncountably infinite number of possible values in an interval
(e.g. x can take all values in the interval [a, b])
◼ example (Lecture 4): amount of time (seconds) workers on
an assembly line take to complete a particular task → x can
take all non-negative real numbers
4. Lect05 Continuous Probability Distributions 4
Continuous random variables
– cont. – Sect.8.1
◼ Discrete random variables:
◼ all possible values of random variable X can be listed
◼ it is meaningful to consider probability P(X = x)
◼ Continous random variables:
◼ all possible values cannot be listed: there is always another
possible value between any two of its values
◼ probability that a continuous random variable X will assume
any particular value is zero: P(X = x) = 0 !
◼ the only meaningful events for a continuous random variable
are intervals
◼ comparable situation: a line segment has a positive length,
while no single point on the line segment does
5. Lect05 Continuous Probability Distributions 5
Probability density functions
– Sect.8.1
◼ The probability that X falls between a
and b is found by calculating the area
under the graph of f(x) between a and b:
Area = 1
f(x)
=
=
=
b
x
a
x
dx
x
f
b
X
a
P )
(
)
(
a b
f(x)
◼ To calculate probabilities we define a
probability density function f(x)
◼ The density function satisfies the
following conditions:
◼ f(x) is non-negative
◼ the total area under the curve
representing f(x) equals 1
Important: f(x)
P(X = x)
6. Lect05 Continuous Probability Distributions 6
Uniform distribution –
continuous case – Sect.8.1
◼ obtained as follows:
◼ (Continuous) uniform distribution:
◼ example of continuous random variable (r.v.)
◼ domain of continuous uniform r.v. X: a x b
◼ probability density function (p.d.f.):
◼ hence, outcomes are distributed evenly
◼ expected value and variance:
x b
1
f(x) a
b a
=
−
2
(b a)
a b
E(X) V(X)
2 12
−
+
= =
( )
x b x b 2
x a x a
E(X) xf(x)dx V(X) x f(x)dx
= =
= =
= = −
7. Lect05 Continuous Probability Distributions 7
Uniform distribution
– example – Sect.8.1
◼ Solution:
◼ Example (= example 7.1 5th ed.):
◼ the time elapsed between the placement of an order and the
delivery time (X, measured in minutes) is uniformly
distributed between 100 and 180 minutes
◼ define the density function and draw the graph
◼ what proportion of orders takes between 2 and 2.5 hours to
deliver:
100 180
1/80
x
120 150
P(120 X 150) = (150-120)(1/80) = .375
?
)
150
120
( =
X
P
f(x) = 1/80 100 x 180 probability density function
8. Lect05 Continuous Probability Distributions 8
Normal distribution
– Sect.8.2
◼ Most important continuous distribution in statistics:
◼ many random variables can be properly modeled as normally
distributed
◼ many distributions can be approximated by a normal
distribution
◼ the normal distribution is the cornerstone distribution in
statistical inference; the (asymptotic) distribution of sample
means is normal (see next lecture)
9. Lect05 Continuous Probability Distributions 9
Normal distribution
– definition – Sect.8.2
◼ Notation:
◼ Normal distribution:
◼ a random variable X is said to have a normal
distribution with parameters μ and σ if its p.d.f. is:
2
)
(
)
(
=
= X
V
X
E
...
71828
.
2
and
...
14159
.
3
with
2
1
exp
2
1
)
(
2
=
=
−
−
−
=
e
x
x
x
f
◼ Also:
( )
,
~ N
X
10. Lect05 Continuous Probability Distributions 10
Normal distribution
– shape – Sect.8.2
◼ The p.d.f. of the normal distribution is bell shaped
and symmetrical around :
σ
μ
11. Lect05 Continuous Probability Distributions 11
Normal distribution
– shape (cont.) – Sect.8.2
How does the standard deviation affect the shape of f(x)?
= 2
=3
=4
= 10 = 11 = 12
How does the expected value affect the location of f(x)?
12. Lect05 Continuous Probability Distributions 12
Standard normal distribution
– Sect.8.2
◼ Two facts help calculate normal probabilities:
◼ the normal distribution is symmetrical
◼ any normal distribution can be transformed into a specific
normal distribution, called standard normal distribution, with
mean 0 and variance 1
◼ Example:
◼ the time it takes to complete a standard entrance exam is
assumed to be normally distributed, with a mean of 60
minutes and a standard deviation of 8 minutes
◼ what is the probability that a student will complete the exam
within 70 minutes?
13. Lect05 Continuous Probability Distributions 13
Standard normal distribution
– cont. – Sect.8.2
◼ Solution:
◼ if X denotes the time taken to complete the exam, we look
for the probability P(X < 70) where μ = 60 and σ = 8
◼ this probability can be calculated by creating a new normal
variable, the standard normal variable (Z):
E(Z) = 0
V(Z) = 1
Z measures the number of standard deviation
units (σ) that X deviates from its mean (μ)
Therefore, once probabilities for Z are calculated,
probabilities of any normal variable can be found
Every normal variable X with some μ
and σ, can be transformed into this Z
X
Z
−
=
14. Lect05 Continuous Probability Distributions 14
Standard normal distribution
– cont. – Sect.8.2
◼ To complete the calculation we need to compute the
standard normal probability P(Z < 1.25)
◼ Solution (cont.):
P(X 70) = P(X – 70 – 60)
= P(Z 1.25)
X 70 60
P
8
− −
=
15. Lect05 Continuous Probability Distributions 15
Standard normal table
– Table 3 App. B – Sect.8.2
◼ Solution (cont.):
◼ standard normal probabilities
have been calculated and are
provided in a table:
The tabulated
probabilities
correspond to the
area between Z = −
and some Z = z0
z0
0
P(Z < z0)
Table 3 - Appendix B (pages B-8 and B-9)
z 0.03 0.04 0.05 0.06 0.07
1.0 0.8485 0.8508 0.8531 0.8554 0.8577
1.1 0.8708 0.8729 0.8749 0.8770 0.8790
1.2 0.8907 0.8925 0.8944 0.8962 0.8980
1.3 0.9082 0.9099 0.9115 0.9131 0.9147
1.4 0.9236 0.9251 0.9265 0.9279 0.9292
16. Lect05 Continuous Probability Distributions 16
Standard normal table
– Table 3 App. B – Sect.8.2
◼ Solution (cont.):
z0
0
Table 3 - Appendix B (pages B-8 and B-9)
z 0.03 0.04 0.05 0.06 0.07
1.0 0.8485 0.8508 0.8531 0.8554 0.8577
1.1 0.8708 0.8729 0.8749 0.8770 0.8790
1.2 0.8907 0.8925 0.8944 0.8962 0.8980
1.3 0.9082 0.9099 0.9115 0.9131 0.9147
1.4 0.9236 0.9251 0.9265 0.9279 0.9292
In this example
z0 = 1.25
P(X 70) = P(X – 70 – 60)
= P(Z 1.25)
X 70 60
P
8
− −
=
= 0.8944
0.8944
17. Lect05 Continuous Probability Distributions 17
−1.25 z
0 1.25
Calculating normal
probabilities – Sect.8.2
◼ Probabilities of the type P(Z > z0) can easily be
derived from Table 3 by applying the complement rule
(see Lecture 3), hence:
◼ P(Z>1.25) = 1 – P(Z<1.25) = 1 – 0.8944 = 0.1056
◼ From the symmetry of the normal distribution it
follows that P(Z < –1.25) = P(Z > 1.25) = 0.1056...
◼ …as also follows from Table 3
◼ Hence, P(Z < –z) + P(Z < z) = 1
for any z (verify yourself in Table 3!)
Table 3 - Appendix B (pages B-8 and B-9)
z 0.03 0.04 0.05 0.06 0.07
-1.4 0.0764 0.0749 0.0735 0.0721 0.0708
-1.3 0.0918 0.0901 0.0885 0.0869 0.0853
-1.2 0.1093 0.1075 0.1056 0.1038 0.1020
-1.1 0.1292 0.1271 0.1251 0.1230 0.1210
-1.0 0.1515 0.1492 0.1469 0.1446 0.1423
0.1056
0.1056
0.1056
P(Z < −z) = P(Z > +z)
18. Lect05 Continuous Probability Distributions 18
Calculating normal prob.
– examples – Sect.8.2
◼ (a) find the probability P(Z > 1.47)
P(Z > 1.47) = 1 – P(Z < 1.47) = 1 – 0.9292 = 0.0708
Tip: always first
sketch the
distribution and
the relevant area !
1.47 z
0
0.9292
0.0708
19. Lect05 Continuous Probability Distributions 19
Calculating normal prob.
– examples – Sect.8.2
◼ (b) find the probability P(−2.25 < Z < 1.85)
Hence, P(−2.25 < Z < 1.85) = 0.9678 − 0.0122 = 0.9556
P(Z < −2.25) = 0.0122 P(Z < 1.85) = 0.9678
−2.25 z
0 1.85
P(−2.25 < Z < 1.85) = P(Z < 1.85) – P(Z < –2.25)
0.9556
20. Lect05 Continuous Probability Distributions 20
Normal distribution
– applications – Sect.8.2
◼ Example (= example 7.3 5th ed.):
◼ the rate of return (X) on an investment is normally distributed
with mean of 30% and standard deviation of 10%
◼ what is the probability that the return will exceed 55%?
0 z = 2.5
=1 – P(Z < 2.5) = 1 – 0.9938 = 0.0062
µ = 30% x = 55%
P(X > 55) = P(Z > ) = P(Z > 2.5)
55 - 30
10
0.9938
0.0062
0.0062 0.9938
21. Lect05 Continuous Probability Distributions 21
P(X < 22) = P(Z < )
22 30
10
−
Normal distribution
– applications – Sect.8.2
◼ Example (cont.):
◼ what is the probability that the return will be less than 22%?
30% x
22%
0 z
−0.8
= P(Z < −0.8) = 0.2119
0.2119
0.2119
22. Lect05 Continuous Probability Distributions 22
Normal distribution
– applications – Sect.8.2
◼ Example (= example 7.4 5th ed.):
◼ if Z is a standard normal variable, determine the value z for
which P(Z < z) = .6331
◼ → use Table 3 backward:
since P(Z < 0.34) = 0.6331
Table 3 - Appendix B (pages B-8 and B-9)
z 0.02 0.03 0.04 0.05 0.06
0.1 0.5478 0.5517 0.5557 0.5596 0.5636
0.2 0.5871 0.5910 0.5948 0.5987 0.6026
0.3 0.6255 0.6293 0.6331 0.6368 0.6406
0.4 0.6628 0.6664 0.6700 0.6736 0.6772
0.5 0.6985 0.7019 0.7054 0.7088 0.7123
0.6331
0.6331
0 z
z = 0.34
23. Lect05 Continuous Probability Distributions 23
Table 3 - Appendix B (pages B-8 and B-9)
z 0.04 0.05 0.06 0.07 0.08
1.7 0.9591 0.9599 0.9608 0.9616 0.9625
1.8 0.9671 0.9678 0.9686 0.9693 0.9699
1.9 0.9738 0.9744 0.9750 0.9756 0.9761
2.0 0.9793 0.9798 0.9803 0.9808 0.9812
2.1 0.9838 0.9842 0.9846 0.9850 0.9854
Normal distribution
– applications – Sect.8.2
◼ Example (= example 7.5 5th ed.): determine z0.025 (critical
value)
◼ Solution: the value zA is defined as the z-value for which the
area to the right(!) of zA under the standard normal curve is A:
P(Z > zA) = A (right-tail probability)
=1.96
-1.96= −z0.025
0.025
Table 3: P(Z < 1.96) = 0.975
→ z0.025 = 1.96
1.9
0.06
0.9750
P(Z < z0.025) = 0.975 (= 1 – A)
z0.025
0
P(Z > z0.025) = 0.025 (= A)
24. Lect05 Continuous Probability Distributions 24
Normal distribution
– properties – Sect.8.2
and
◼ Special case: X (= X1) and Y (= X2) are identically
distributed:
and X and Y are independent random variables
then
◼ If
and a = b = 1/2, then it follows that
(X1 + X2)/2 is sample
mean of X1 and X2
Verify
yourself !
( ) ( )
X X Y Y
X ~ N , and Y ~ N ,
( )
2 2 2 2
X Y X Y
aX bY ~ N a b , a b
+ = + = +
( )
2 2 2 2
X Y X Y
aX bY ~ N a b , a b
− = − = +
( )
1 2 X x
X , X ~ N ,
1 2 x
X
X X
~ N ,
2 2
+
= =
25. Lect05 Continuous Probability Distributions 25
Exponential distribution
– Sect.8.3
◼ The exponential distribution is frequently applied
when interested in elapsed time intervals
◼ Examples:
◼ the length of time between telephone calls
◼ the length of time between arrivals at a service station
◼ the life-time of electronic components
◼ When the number of occurrences of an event has a
Poisson distribution with average µ per unit of time,
the time between the occurrences has an exponential
distribution with average 1/µ time unit
26. Lect05 Continuous Probability Distributions 26
Exponential distribution
– definition – Sect.8.3
◼ Notation:
◼ A random variable X is said to be exponentially
distribution with parameter if its p.d.f. is:
◼ The expected value and the variance of X are:
( ) x
f x e x 0
where 0
−
=
( ) ( ) 2
1 1
E X V X
= =
( )
X ~ Exp
27. Lect05 Continuous Probability Distributions 27
( ) ( )
( )
t a
t a
t t
t 0
t 0
a
P X a 1 P 0 X a
1 e dt 1 e
e
=
=
− −
=
=
−
= −
= − = +
=
Exponential distribution
– definition – Sect.8.3
◼ Also, the probability that an exponentially distributed
r.v. X will take a value greater than a specified
nonnegative number a can be easily computed using:
28. Lect05 Continuous Probability Distributions 28
Exponential distribution
– shape – Sect.8.3
Exponential distribution
for = 0.5, 1, 2
0
0.5
1
1.5
2
2.5
f(x) = 2e -2x
f(x) = 1e -1x
f(x) = 0.5e -0.5x
0 1 2 3 4 5
0
0.5
1
1.5
2
2.5
0 a b
P(a < X < b) =
= P(X > a) − P(X > b)
= e−a - e−b
29. Lect05 Continuous Probability Distributions 29
Exponential distribution
– application – Sect.8.3
◼ Example (= example 7.6 5th ed.; see also example 6.11 5th ed.
mentioned in Lecture 4):
◼ cars arrive randomly and independently at a tollbooth at an
average of 360 cars per hour
◼ let Y = number of arrivals per minute, then we know already
from Lecture 4 that Y ~ Poisson( = 6) [average arrival rate
is 360/60 = 6 cars per minute]
◼ (a) use the exponential distribution to find the probability
that the next car will not arrive within half a minute
◼ (b) use the Poisson distribution to find the probability that
no car will arrive within the next half minute
30. Lect05 Continuous Probability Distributions 30
Exponential distribution
– application – Sect.8.3
◼ Solution (a):
◼ let X = time (in minutes) that elapses before the next car
arrives, then the desired probability is P(X > 0.5), i.e. it
takes more than 0.5 minute before next car arrives
◼ if cars arrive at an average rate of 6 cars per minute, then
the average inter-arrival time is 1/6 minute [= E(X) = 1/],
hence X ~ Exp( = 6)
◼ it follows that P(X > 0.5) = e−60.5 = e−3 = 0.0498
◼ note that the parameter of the Poisson variable Y ( = 6)
always equals the parameter of the corresponding
exponential variable X ( = 6), when same units of time
(here: minutes) are used
31. Lect05 Continuous Probability Distributions 31
Exponential distribution
– application – Sect.8.3
◼ Solution (b):
◼ now we use Y = number of arrivals per minute
◼ we know that Y ~ Poisson( = 6 per minute), or Y ~
Poisson( = 3 per half minute)
◼ desired probability P(Y = 0 (per half minute)):
Comment: If the first car will not arrive within the next half minute
then no car will arrive within the next half minute. Therefore, not
surprisingly, the probability found here is the exactly same
probability found in the previous question
( )
3 0
3
e 3
P Y 0 e 0.0498
0!
−
−
= = = =
32. Lect05 Continuous Probability Distributions 32
Exponential distribution
– application – Sect.8.3
◼ Example (= example 7.7 5th ed.):
◼ the lifetime of a transistor is exponentially distributed, with a
mean of 1,000 hours
◼ what is the probability that the transistor will last between
1,000 and 1,500 hours
◼ Solution:
◼ let X denote the lifetime of a transistor (in hours)
◼ E(X) =1000 = 1/, so = 1/1000 = 0.001
◼ P(1000 < X < 1500) = e−0.0011000 − e−0.0011500 = 0.1448