Chapter 5 Expectations
description
Transcript of Chapter 5 Expectations
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Chapter 5 Expectations:
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ContentIntroductionExpectation of a Function of a Random VariableExpectation of Functions of Multiple Random VariablesImportant Properties of ExpectationConditional ExpectationsMoment Generating FunctionsInequalitiesThe Weak Law of Large Numbers and Central Limit Theorems
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Chapter 5 ExpectationsIntroduction
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Definition ExpectationThe expectation (mean), E[X] or X, of a random variable X is defined by:
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Definition ExpectationThe expectation (mean), E[X] or X, of a random variable X is defined by:provided that the relevant sum or integral is absolutely convergent, i.e.,
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Definition ExpectationThe expectation (mean), E[X] or X, of a random variable X is defined by:provided that the relevant sum or integral is absolutely convergent, i.e.,
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Example 1Let X denote #defectives in the experiment.
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Example 2
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Example 3pdf
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Example 3
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Chapter 5 ExpectationsExpectation of a Function of a Random Variable
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The Expectation of Y=g(X)
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The Expectation of Y=g(X)
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Example 4
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Example 5
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Moments
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X :
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Example 6X ~ B(n, p)E[X]=? Var[X]=?
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Example 6X ~ B(n, p)E[X]=? Var[X]=?
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Example 6X ~ B(n, p)E[X]=? Var[X]=?
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Example 7X ~ Exp()E[X]=? Var[X]=?
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Summary of Important Moments of Random Variables
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Chapter 5 ExpectationsExpectation of Functions of Multiple Random Variables
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The Expectation of Y = g(X1, , Xn)
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Example 8p(x, y)
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Example 9
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Chapter 5 ExpectationsImportant Properties of Expectation
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LinearityE1.E2.X1, X2, , Xn
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Example 10XYE[X+Y] = E[X]+E[Y].
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A QuestionXYE[X+Y] = E[X]+E[Y].?
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IndependenceE3.If random variables X1, . . ., Xn are independent, then
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Example 11XYE[XY] = E[X]E[Y].
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A QuestionXYE[XY] = E[X]E[Y].?
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Example 12
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A Question?
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The Variance of SumDefine
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The Variance of Sum
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The Covariance
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The Covariance
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Example 13
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A Question?
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Properties Related to CovarianceE4.E5.
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Properties Related to CovarianceE4.E5.Fact:
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Properties Related to CovarianceE4.E5.E6.E7.
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Example 14
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Example 14
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More Properties on CovarianceE8.
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More Properties on CovarianceE8.E9.
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Example 16
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Example 16
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Example 16
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Theorem 1 Schwartz Inequality
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Theorem 1 Schwartz InequalityPf)
E=*E
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Theorem 1 Schwartz InequalityPf)
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Theorem 1 Schwartz InequalityPf)
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CorollaryE10.Pf)
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Correlation CoefficientE11.
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Correlation CoefficientE11.Fact:Is the converse also true?
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Correlation CoefficientE11.E12.Pf)
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Example 18
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Example 18
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Example 18
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Example 19
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Example 19Method 1:
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Example 19Method 2:Facts:
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Chapter 5 ExpectationsConditional Expectations
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Definition Conditional Expectations
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Facts a function of X (x)See text for the proofE13.
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Conditional Variances
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Example 20
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Chapter 5 ExpectationsMoment Generating Functions
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Moment Generating FunctionsMomentsMoments
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Moment Generating FunctionsThe moment generating function MX(t) of a random variable X is defined byThe domain of MX(t) is all real numbers such that eXt has finite expectation.
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Example 21
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Example 22
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Summary of Important Moments of Random Variables
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Moment Generating FunctionsThe moment generating function MX(t) of a random variable X is defined byThe domain of MX(t) is all real numbers such that eXt has finite expectation.MX(t) ?
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Moment Generating Functions
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Moment Generating Functions
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Moment Generating Functions
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Moment Generating Functions
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Example 23
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Example 23
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Example 23
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Example 23
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Example 23
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Correspondence or Uniqueness TheoremLet X1, X2 be two random variables.
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Example 24
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Example 24
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Example 24
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Example 24
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Example 24
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Theorem Linear TranslationPf)
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Theorem ConvolutionPf)
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Example 25
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Example 25
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Example 25
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Example 25
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Example 25
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Example 26
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Example 26
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Example 26
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Theorem of Random Variables Sum
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Theorem of Random Variables SumWe have proved the above five using probability generating functions.They can also be proved using moment generating functions.
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Theorem of Random Variables Sum
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Theorem of Random Variables Sum
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Theorem of Random Variables Sum
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Theorem of Random Variables Sum
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Theorem of Random Variables Sum
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Theorem of Random Variables Sum
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Theorem of Random Variables Sum
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Theorem of Random Variables Sum
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Chapter 5 ExpectationsInequalities
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Theorem Markov InequalityLet X be a nonnegative random variable with E[X] = .Then, for any t > 0,
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Theorem Markov InequalityDefineWhy?
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Theorem Markov InequalityDefine
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Example 27MTTF Mean Time To Failure
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Example 27MTTF Mean Time To FailureBy MarkovBy Exponential Distribution
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Theorem Chebyshev's Inequality
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Theorem Chebyshev's Inequality
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Theorem Chebyshev's InequalityFacts:
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Theorem Chebyshev's InequalityFacts:
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Example 28
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Example 28
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Chapter 5 ExpectationsThe Weak Law of Large Numbers andCentral Limit Theorems
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The Parameters of a PopulationWe may never have the chance to know the values of parameters in a population exactly.
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Sample Meaniid random variables
iid: identical independent distributions Sample Mean
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Expectation & Variance of
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Expectation & Variance of
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Expectation & Variance ofn?
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Theorem Weak Law of Large NumbersLet X1, , Xn be iid random variables having finite mean .
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Theorem Weak Law of Large NumbersLet X1, , Xn be iid random variables having finite mean .Chebyshev's Inequality
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Central Limit TheoremLet X1, , Xn be iid random variables having finite mean and finite nonzero variance 2.
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Central Limit TheoremLet X1, , Xn be iid random variables having finite mean and finite nonzero variance 2.
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Central Limit Theorem
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Central Limit Theorem
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Central Limit Theorem
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Central Limit Theoremn0
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Central Limit Theoremn
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Central Limit Theorem
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Central Limit Theorem
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Central Limit TheoremLet X1, , Xn be iid random variables having finite mean and finite nonzero variance 2.
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Normal ApproximationBy the central limit theorem, when a sample size is sufficiently large (n > 30), we can use normal distribution to approximate certain probabilities regarding to the sample or the parameters of its corresponding population.
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Example 29Let Xi represent the lifetime of ith bulbWe want to findn > 30
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Example 30n > 30
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Example 30
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