Probability
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3.1 Extreme Values of Functions
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
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Monte Carlo methods rely on repeated random sampling to compute results. They generate random samples from a population according to a probability distribution and use them to obtain numerical results. The founders of the Monte Carlo method were J. von Neumann and S. Ulam during the Manhattan Project in the 1940s. Monte Carlo methods can be used to solve multidimensional integrals and have better convergence than classical numerical integration methods for dimensions greater than 4. The variance of Monte Carlo estimates decreases as 1/N, where N is the number of samples, resulting in slow convergence. Variance reduction techniques can improve the convergence rate.
Lesson 4.1 Extreme Values
1) A function has an absolute maximum value on its domain if its value is greater than or equal to its value at all other points in its domain, and an absolute minimum value if its value is less than or equal to its value at all other points.
2) By the Extreme Value Theorem, if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value within that interval. These extreme values can occur at interior points or endpoints.
3) A local extreme value of a function is a maximum or minimum value within a neighborhood of some interior point, where the function's derivative is equal to 0 or undefined at that point, according to the Local Extreme Value Theorem.
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- The version space is the set of all hypotheses consistent with the training data.
- Controlling the complexity of the hypothesis class H via measures like VC dimension can improve generalization.
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Lesson 4.1 Extreme Values
1) A function has an absolute maximum value on its domain if its value is greater than or equal to its value at all other points in its domain, and an absolute minimum value if its value is less than or equal to its value at all other points.
2) By the Extreme Value Theorem, if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value within that interval. These extreme values can occur at interior points or endpoints.
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Simulation involves imitating the performance of a system using a model to generate sample experiments without observing the real system. Monte Carlo simulation specifically involves an element of chance and is useful when direct experimentation is impossible or too costly. It generates random numbers that can represent values of random variables to simulate observations. Common techniques include using uniform random numbers between 0 and 1 to represent distributions and the Box-Muller method for generating normal random variables.
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Monte Carlo methods use random sampling to solve problems numerically. They work by setting up probabilistic models and running simulations using random numbers. This allows approximating solutions to problems in physics, finance, optimization, and other fields. Examples include estimating pi by simulating dart throws, and using a "drunken wino" random walk simulation to approximate the solution to a partial differential equation on a grid. The accuracy of Monte Carlo methods increases with more simulation iterations, requiring truly random numbers for best results.
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Monte Carlo simulation is a statistical technique that uses random numbers and probability to simulate real-world processes. It was developed in the 1940s by scientists working on nuclear weapons research. Monte Carlo simulation provides approximate solutions to problems by running simulations many times. It allows for sensitivity analysis and scenario analysis. Some examples include estimating pi by randomly generating points within a circle, and approximating integrals by treating the area under a curve as a target for random darts. The technique provides probabilistic results and allows modeling of correlated inputs.
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- Events: One or more possible outcomes of an experiment.
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- Mutually exclusive events: Events that cannot occur together.
- Exhaustive events: A set of events where one of the events must occur.
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- Examples of calculating probability for single-step experiments like rolling a die
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- The relationship between experimental probability from trials and theoretical probability as the number of trials increases
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This document discusses the concept of probability. It defines probability as a measure of how likely an event is to occur. Probabilities can be described using terms like certain, likely, unlikely, and impossible. Mathematically, probabilities are often expressed as fractions, with the numerator representing the number of possible outcomes for an event and the denominator representing the total number of possible outcomes. The document provides examples to illustrate concepts like independent and conditional probabilities, as well as complementary events and the gambler's fallacy.
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There are two ways to count the number of possible outcomes of an experiment:
1) Using a tree diagram to list out all the combinations
2) Using the Fundamental Counting Principle, which involves multiplying the number of choices for each event together.
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Monte Carlo simulation is a statistical technique that uses random numbers and probability to simulate real-world processes. It was developed in the 1940s by scientists working on nuclear weapons research. Monte Carlo simulation provides approximate solutions to problems by running simulations many times. It allows for sensitivity analysis and scenario analysis. Some examples include estimating pi by randomly generating points within a circle, and approximating integrals by treating the area under a curve as a target for random darts. The technique provides probabilistic results and allows modeling of correlated inputs.
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Algorithm and Flowchart.
Or ppt,new
This document discusses methods of simulation and Monte Carlo simulation. It is authored by a group including Roy Thomas, Sam Scaria, Sonu Sebastian, and others. The document defines simulation as using a model of a real system to conduct experiments on a computer in order to describe, explain, and predict the behavior of the real system. Monte Carlo simulation is described as using probability and sampling to solve complicated equations. Key steps of Monte Carlo simulation include drawing a flow diagram, determining variable distributions, selecting random numbers, and applying mathematical functions to obtain solutions. Examples of applications include queuing problems, inventory problems, and risk analysis.
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Selected items from the set theory and methodology and philosophy of mathematics and computer programming.
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The Monte Carlo method uses random numbers and probability statistics to solve complex problems through approximation. It was developed in the 1940s by physicists working on nuclear weapons who needed to model radiation shielding. The method involves defining a domain of possible inputs, randomly generating inputs from that domain, performing deterministic computations using the inputs, and aggregating the results. Monte Carlo methods are useful when problems cannot be solved analytically due to uncertainty. They are commonly used to value options and other financial derivatives.
Iterative1
The document describes the iterative method for finding roots of equations. It discusses:
1) How to set up an iterative process to find a root by taking an initial approximation x0 and repeatedly applying the function pi(x) to get closer approximations x1, x2, etc. until the change between successive values is very small.
2) An example of using the method to find the root of the quadratic equation 2x^2 - 4x + 1 = 0. It shows calculating successive approximations that converge to the root.
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The document discusses the history and development of Monte Carlo simulation methods in financial engineering. Some key points:
1) Monte Carlo simulation techniques originated from games of chance and probabilistic concepts in the 17th century. They were later applied to calculating integrals and solving differential equations.
2) In the 1940s/50s, the techniques were developed and applied at Los Alamos National Laboratory, coining the term "Monte Carlo."
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The monkey can move one space at a time in any cardinal direction on a planar grid. For any point on the grid, if the sum of the digits of the absolute value of the x coordinate plus the sum of the digits of the absolute value of the y coordinate is less than or equal to 19, the monkey can access that point. Starting from (0,0), the question asks how many points on the grid the monkey can access given this rule.
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The document discusses how to determine if a function is increasing or decreasing based on the sign of its derivative, and provides an example of finding the intervals where a function is increasing or decreasing. It also discusses how to find and classify critical points of a function by taking the derivative, setting it equal to 0, and using the first derivative test to determine if the critical points are local maxima or minima.
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This document defines key terminology related to probability, including:
- Sample space: The set of all possible outcomes of a random experiment.
- Events: One or more possible outcomes of an experiment.
- Equally likely events: Events with an equal chance of occurring.
- Mutually exclusive events: Events that cannot occur together.
- Exhaustive events: A set of events where one of the events must occur.
It also provides examples and introduces the classical definition of probability as the number of favorable outcomes divided by the total number of outcomes.
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This document provides an introduction to simple probability concepts including:
- Definitions of outcomes, favorable outcomes, and theoretical probability
- Examples of calculating probability for single-step experiments like rolling a die
- Representing two-step experiments using ordered pairs and calculating probabilities using tables
- The relationship between experimental probability from trials and theoretical probability as the number of trials increases
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This document discusses the concept of probability. It defines probability as a measure of how likely an event is to occur. Probabilities can be described using terms like certain, likely, unlikely, and impossible. Mathematically, probabilities are often expressed as fractions, with the numerator representing the number of possible outcomes for an event and the denominator representing the total number of possible outcomes. The document provides examples to illustrate concepts like independent and conditional probabilities, as well as complementary events and the gambler's fallacy.
Probability powerpoint
The document discusses probability and introduces several online activities and games for students to learn about probability. Students are instructed to visit specific pages on BrainPop, Maths Online, Interactivate: Activities, Skillswise: Numbers, Johnnie's Math Page, and take an online quiz. After completing the activities, students are asked to write a paragraph about what they learned about probability and come up with their own probability experiment.
Probability Overview
There are two ways to count the number of possible outcomes of an experiment:
1) Using a tree diagram to list out all the combinations
2) Using the Fundamental Counting Principle, which involves multiplying the number of choices for each event together.
To calculate the probability of compound events (events made up of two or more simple events), you first determine if the events are independent or dependent. For independent events, the probability of one event does not affect the other event. You calculate the probability by multiplying the individual probabilities together.
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The document defines probability as the ratio of desired outcomes to total outcomes. It provides examples of calculating probabilities of outcomes from rolling a die or flipping a coin. It explains that probabilities of all outcomes must sum to 1. It also discusses calculating probabilities of multiple events using "and" or "or", and defines experimental probability as the ratio of outcomes to trials from an experiment.
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1) The document provides an introduction to basic probability concepts such as sample space, events, simple and compound events, favorable outcomes, and calculating probability as a ratio, fraction, or percentage.
2) It gives examples of calculating probabilities for simple events like flipping a coin or rolling a die, as well as compound events involving multiple steps.
3) Key points are that the probability of any event is between 0 and 1, and the probability of the complimentary event of E is 1 - the probability of E.
Probability
This document provides an overview of key concepts in probability, including experiment, event, sample space, unions and intersections of events, mutually exclusive events, and methods of assigning probabilities. It discusses experiments, events, and sample spaces. It defines union and intersection of sets. It covers classical, relative frequency, and subjective probabilities. It also discusses rules for counting sample points, including multiplication, permutation, and combination. Examples are provided to illustrate calculating probabilities.
Sfs4e ppt 05
This document contains a chapter about probability from a Pearson textbook. It includes objectives, definitions, examples, and explanations about key concepts in probability such as:
- The addition rule for disjoint (mutually exclusive) events, which states that the probability of event A or B occurring is equal to the sum of the individual probabilities if A and B cannot both occur.
- Subjective probabilities, which are based on personal judgment rather than experimental data.
- Using simulations and empirical methods to estimate probabilities based on observed frequencies from repeated experiments.
Pre-Cal 40S May 27, 2009
The document provides an introduction to basic probability concepts including:
- Sample space is the set of all possible outcomes of an experiment.
- An event is a subset of outcomes within the sample space.
- Probability is expressed as a ratio of favorable outcomes to total outcomes.
- Probability ranges from 0 to 1, with 0 being impossible and 1 being certain.
- Complementary events have probabilities that sum to 1.
7-Experiment, Outcome and Sample Space.pptx
1) 1.56 and 1.0 cannot be probabilities because probabilities must be between 0 and 1.
2) 0.46, 0.09, 0.96, 0.25, 0.02 can be probabilities because they are between 0 and 1.
3) a) The probability of obtaining a number less than 4 is 3/6 = 1/2
b) The probability of obtaining a number between 3 and 6 is 4/6 = 2/3
Pre-Cal 40S Slides May 21, 2008
The document provides an introduction to probability concepts including sample space, events, simple and compound events, calculating probabilities as fractions or decimals, certain, impossible, and complementary events. It gives examples of determining the sample space and probabilities for experiments involving rolling dice, flipping coins, and drawing cards. It poses questions about identifying sample spaces and calculating probabilities for experiments combining dice rolls, coin flips, and bus and train arrival times.
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- 1. Now it’s time to look at… Discrete Probability CMSC 203 Discrete Structures 1
- 2. Discrete Probability Everything you have learned about counting constitutes the basis for computing the probability of events to happen. In the following, we will use the notion experiment for a procedure that yields one of a given set of possible outcomes. This set of possible outcomes is called the sample space of the experiment. An event is a subset of the sample space. CMSC 203 Discrete Structures 2
- 3. Discrete Probability If all outcomes in the sample space are equally likely, the following definition of probability applies: The probability of an event E, which is a subset of a finite sample space S of equally likely outcomes, is given by p(E) = |E|/|S|. Probability values range from 0 (for an event that will never happen) to 1 (for an event that will always happen whenever the experiment is carried out). CMSC 203 Discrete Structures 3
- 4. Discrete Probability Example I: An urn contains four blue balls and five red balls. What is the probability that a ball chosen from the urn is blue? Solution: There are nine possible outcomes, and the event “blue ball is chosen” comprises four of these outcomes. Therefore, the probability of this event is 4/9 or approximately 44.44%. CMSC 203 Discrete Structures 4
- 5. Discrete Probability Example II: What is the probability of winning the lottery 6/49, that is, picking the correct set of six numbers out of 49? Solution: There are C(49, 6) possible outcomes. Only one of these outcomes will actually make us win the lottery. p(E) = 1/C(49, 6) = 1/13,983,816 CMSC 203 Discrete Structures 5
- 6. Complimentary Events Let E be an event in a sample space S. The probability of an event –E, the complimentary event of E, is given by p(-E) = 1 – p(E). This can easily be shown: p(-E) = (|S| - |E|)/|S| = 1 - |E|/|S| = 1 – p(E). This rule is useful if it is easier to determine the probability of the complimentary event than the probability of the event itself. CMSC 203 Discrete Structures 6
- 7. Complimentary Events Example I: A sequence of 10 bits is randomly generated. What is the probability that at least one of these bits is zero? Solution: There are 210 = 1024 possible outcomes of generating such a sequence. The event –E, “none of the bits is zero”, includes only one of these outcomes, namely the sequence 1111111111. Therefore, p(-E) = 1/1024. Now p(E) can easily be computed as p(E) = 1 – p(-E) = 1 – 1/1024 = 1023/1024. CMSC 203 Discrete Structures 7
- 8. Complimentary Events Example II: What is the probability that at least two out of 36 people have the same birthday? Solution: The sample space S encompasses all possibilities for the birthdays of the 36 people, so |S| = 36536. Let us consider the event –E (“no two people out of 36 have the same birthday”). –E includes P(365, 36) outcomes (365 possibilities for the first person’s birthday, 364 for the second, and so on). Then p(-E) = P(365, 36)/36536 = 0.168, so p(E) = 0.832 or 83.2% CMSC 203 Discrete Structures 8