Chapter 5 Expectations

Chapter 5 Expectations:

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

Chapter 5 ExpectationsIntroduction

Definition ExpectationThe expectation (mean), E[X] or X, of a random variable X is defined by:

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.,

Example 1Let X denote #defectives in the experiment.

Example 2

Example 3pdf

Example 3

Chapter 5 ExpectationsExpectation of a Function of a Random Variable

The Expectation of Y=g(X)

Example 4

Example 5

Moments

Example 6X ~ B(n, p)E[X]=? Var[X]=?

Example 7X ~ Exp()E[X]=? Var[X]=?

Summary of Important Moments of Random Variables

Chapter 5 ExpectationsExpectation of Functions of Multiple Random Variables

The Expectation of Y = g(X1, , Xn)

Example 8p(x, y)

Example 9

Chapter 5 ExpectationsImportant Properties of Expectation

LinearityE1.E2.X1, X2, , Xn

Example 10XYE[X+Y] = E[X]+E[Y].

A QuestionXYE[X+Y] = E[X]+E[Y].?

IndependenceE3.If random variables X1, . . ., Xn are independent, then

Example 11XYE[XY] = E[X]E[Y].

A QuestionXYE[XY] = E[X]E[Y].?

Example 12

A Question?

The Variance of SumDefine

The Variance of Sum

The Covariance

Example 13

A Question?

Properties Related to CovarianceE4.E5.

Properties Related to CovarianceE4.E5.Fact:

Properties Related to CovarianceE4.E5.E6.E7.

Example 14

More Properties on CovarianceE8.

More Properties on CovarianceE8.E9.

Example 16

Theorem 1 Schwartz Inequality

Theorem 1 Schwartz InequalityPf)

E=*E

Theorem 1 Schwartz InequalityPf)

E

CorollaryE10.Pf)

Correlation CoefficientE11.

Correlation CoefficientE11.Fact:Is the converse also true?

Correlation CoefficientE11.E12.Pf)

0

0

Example 18

Example 19

Example 19Method 1:

Example 19Method 2:Facts:

Chapter 5 ExpectationsConditional Expectations

Definition Conditional Expectations

Facts a function of X (x)See text for the proofE13.

Conditional Variances

Example 20

Chapter 5 ExpectationsMoment Generating Functions

Moment Generating FunctionsMomentsMoments

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.

Example 21

Example 22

Summary of Important Moments of Random Variables

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) ?

Moment Generating Functions

Example 23

Correspondence or Uniqueness TheoremLet X1, X2 be two random variables.

Example 24

Theorem Linear TranslationPf)

Theorem ConvolutionPf)

Example 25

Example 26

Theorem of Random Variables Sum

Theorem of Random Variables SumWe have proved the above five using probability generating functions.They can also be proved using moment generating functions.


?

Chapter 5 ExpectationsInequalities

Theorem Markov InequalityLet X be a nonnegative random variable with E[X] = .Then, for any t > 0,

Theorem Markov InequalityDefineWhy?

Theorem Markov InequalityDefine

Example 27MTTF Mean Time To Failure

Example 27MTTF Mean Time To FailureBy MarkovBy Exponential Distribution

Theorem Chebyshev's Inequality

Theorem Chebyshev's InequalityFacts:

Example 28

Chapter 5 ExpectationsThe Weak Law of Large Numbers andCentral Limit Theorems

The Parameters of a PopulationWe may never have the chance to know the values of parameters in a population exactly.

Sample Meaniid random variables

iid: identical independent distributions Sample Mean

Expectation & Variance of

Expectation & Variance ofn?

Theorem Weak Law of Large NumbersLet X1, , Xn be iid random variables having finite mean .

Theorem Weak Law of Large NumbersLet X1, , Xn be iid random variables having finite mean .Chebyshev's Inequality

Central Limit TheoremLet X1, , Xn be iid random variables having finite mean and finite nonzero variance 2.

Central Limit Theorem


=0 as n

Central Limit Theoremn0

Central Limit Theoremn

Central Limit TheoremLet X1, , Xn be iid random variables having finite mean and finite nonzero variance 2.

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.

Example 29Let Xi represent the lifetime of ith bulbWe want to findn > 30

Example 30n > 30

Example 30

Chart4

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60.00177086040.0026012348

70.0047704810.0058445527

80.01098914370.0119388261

90.02197828740.022172292

100.03861899070.0374367532

110.060185440.057467775

120.083829720.0802027408

130.10501745150.1017636458

140.11894833790.1173909796

150.12234686180.1231162606

160.1147001830.1173909796

170.09831444250.1017636458

180.0772470620.0802027408

190.05575727780.057467775

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230.00668395610.0058445527

240.00322262170.0026012348

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90.02197828740.022172292

100.03861899070.0374367532

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Documents

Chapter 5 Expectations