Lecture 29

Zhihua (Sophia) Su

University of Florida

Mar 25, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Expectation Involving Two Random Variables

STA 4321/5325 Introduction to Probability 2

Expected Values of Functions of RandomVariables

DefinitionIf X, Y are discrete random variables with joint probabilitymass function pX,Y (x, y), then for any functiong(x, y) : X × Y → R, where X = Range(X), and Y =Range(Y ),

E[g(X,Y )] =∑x∈X

∑y∈Y

g(x, y)pX,Y (x, y).

STA 4321/5325 Introduction to Probability 3

Expected Values of Functions of RandomVariables

If the joint probability mass of two discrete random variables(X,Y ) takes positive values in R, i.e.

pX,Y (x, y)

{> 0 if (x, y) ∈ R

= 0 otherwise.

ThenE[g(X,Y )] =

∑(x,y)∈R

g(x, y)pX,Y (x, y).

STA 4321/5325 Introduction to Probability 4

Probabilities Involving Two Random Variables

Of nine executives in a business firm, four are married, threehave never married, and two are divorced. Three of theexecutives are to be selected for promotion. Let Y1 denote thenumber of married executives and Y2 denote the number ofnever-married executives among the three selected forpromotion. Assuming that the three are randomly selected fromthe nine executives.

(a) Find the joint probability function of Y1 and Y2.(b) Find the expected number of married executives among the

three selected for promotion.(c) Find E(Y1Y2).

STA 4321/5325 Introduction to Probability 5

Expected Values of Functions of RandomVariables

DefinitionIf X, Y are continuous random variables with joint probabilitydensity function fX,Y (x, y), then for any functiong(x, y) : R2 → R,

E[g(X,Y )] =

ˆ ∞−∞

ˆ ∞−∞

g(x, y)fX,Y (x, y)dxdy.

STA 4321/5325 Introduction to Probability 6

Expectations Involving Two Random Variables

If two continuous random variables (X,Y ) have joint densityfX,Y (x, y), then

E[g(X,Y )] ≡ˆ ∞−∞

ˆ ∞−∞

g(x, y)fX,Y (x, y)dxdy.

Note again that if

fX,Y (x, y) =

{h(x, y) > 0 if (x, y) ∈ R

0 otherwise,

thenE[g(X,Y )] =

¨

R

g(x, y)h(x, y)dydx.

STA 4321/5325 Introduction to Probability 7

Expectations Involving Two Random Variables

Hence, the same process discussed in last lecture needs to berepeated with R (instead of A ∩R) and g(x, y)h(x, y) instead ofh(x, y).

Example: Suppose that the joint density of (X,Y ) is given by

fX,Y (x, y) =

{e−x if 0 ≤ y ≤ x <∞0 otherwise.

Compute E(Y ) and E(Xe−Y ).

STA 4321/5325 Introduction to Probability 8

Expectations Involving Two Random Variables

You need not always use the order dydx. if it is moreconvenient, you can also use dxdy.

Example: If g(x, y) = x, then using the same joint density as inthe previous example, compute E(X).

STA 4321/5325 Introduction to Probability 9