lecture31.pdf
Transcript of lecture31.pdf
Lecture 31
Zhihua (Sophia) Su
University of Florida
Apr 1, 2015
STA 4321/5325 Introduction to Probability 1
Agenda
Conditional Expectation of Random VariablesProperties Involving Conditional Expectations
Reading assignment: Chapter 5: 5.11
STA 4321/5325 Introduction to Probability 2
Conditional Expectation of Random Variables
Let us recollect that for two random variables X and Y , theconditional distribution of X given Y = y, describes theprobability behavior of the random variables X given theinformation that Y = y. In the same spirit, we define the notionof the conditional expectation of X given Y = y.
STA 4321/5325 Introduction to Probability 3
Conditional Expectation of Random Variables
DefinitionIf X and Y are two discrete random variables with jointprobability mass function pX,Y , then the conditionalexpectation of X given Y = y is defined as:
E(X | Y = y) =∑x∈X
xpX|Y=y(x),
where pX|Y=y(x) is the conditional probability mass of X givenY = y.
STA 4321/5325 Introduction to Probability 4
Conditional Expectation of Random Variables
DefinitionIf X and Y are two continuous random variables with jointprobability density function fX,Y , then the conditionalexpectation of X given Y = y is defined as:
E(X | Y = y) =
ˆ ∞−∞
xfX|Y=y(x)dx,
where fX|Y=y(x) is the conditional probability density of Xgiven Y = y.
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Conditional Expectation of Random Variables
Example: A soft-drink machine has a random supply Y at thebeginning of a given day and dispenses a random amount Xduring the day (with measurements in gallons). It has beenobserved that X and Y have joint density
fX,Y (x, y) =
{12 if 0 ≤ x ≤ y, 0 ≤ y ≤ 2,
0 otherwise.
If it is known that on a particular day 1 gallon was supplied atthe beginning of the day, find the expected value of the amountof soft-drink consumed in that day.
STA 4321/5325 Introduction to Probability 6
Properties Involving Conditional Expectations
Results:Let X and Y be two random variables. Then
E(X) = E[E(X | Y )],
where on the right-hand side, the inside expectation is statedwith respect to the conditional distribution of X given Y , andthe outside expectation is stated with respect to the distributionof Y .
STA 4321/5325 Introduction to Probability 7
Properties Involving Conditional ExpectationsSometimes, the way the distribution of the random variable X isspecified, it is not possible to directly evaluate its expectation,and the previous identity comes is very handy.
Here is an alternative way of evaluating E(X) based on theprevious identity.
1 For every y, find E(X | Y = y).2 Define the function h as h(y) = E(X | Y = y).3 E(X) = E(h(Y )).
Hence, the way to interpret the previous relationship is asfollows.
E(X) = E(h(Y )),where h(y) = E(X | Y = y).
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Properties Involving Conditional Expectations
Example: A quality control plan for an assembly line involvessampling n finished items per day and counting X, the numberof defective items. If p denotes the probability that an item isdefective, then given p, X has a binomial distribution withparameters n and p. However, it is observed that p is a randomquantity, and has a uniform distribution on the interval [0, 14 ].(a) Find E(X).(b) Find SD(X).
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Properties Involving Conditional Expectations
Results:Let X and Y be two random variables. Then
V (X) = E(V (X | Y )) + V (E(X | Y )).
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Properties Involving Conditional Expectations
Here is an algorithm based on this identity to evaluate V (X).1 For every y, find E(X | Y = y) and V (X | Y = y).2 Define the function h as h(y) = E(X | Y = y).3 Define the function h̃ as h̃(y) = V (X | Y = y).4 V (X) = V (h(Y )) + E(h̃(Y )).
Hence, the way to integrate the previous relationship is asfollows:
V (X) = V (h(Y )) + E(h̃(Y )),
where h(y) = E(X | Y = y) and h̃(y) = V (X | Y = y).
STA 4321/5325 Introduction to Probability 11
Properties Involving Conditional Expectations
Note that for any function g,
E(g(X) | Y = y) =
{´∞−∞ g(x)fX|Y=y(x)dx if X is continuous,∑x∈X g(x)pX|Y=y(x) if X is discrete.
STA 4321/5325 Introduction to Probability 12
Properties Involving Conditional Expectations
Example (cont’d)
STA 4321/5325 Introduction to Probability 13