Question

Subject Test Note: - You are attempting question 8 out of 12 Suppose the joint density function of X and Y is defined as, f(x,y) =, osrsys2 Calculate the conditional expectation of X, given that Y=1.5​

Answer 1

Answer: Let X and Y be discrete random variables with joint probability mass function pX,Y (x, y), then the conditional probability mass function was defined in previous lectures as

pX|Y (x|y) = pX,Y (x, y)

pY (y)

,

assuming pY (y) > 0. Let us define

E[X|Y = y] = X

x

xpX|Y (x|y).

ψ(y) = E[X|Y = y] changes with y. The random variable ψ(Y ) is the conditional expectation of X given Y

and denoted as E[X|Y ].

Let X and Y be continuous random variables with joint probability density function fX,Y (x, y). Recall the

conditional probability density function

fX|Y (x|y) = fX,Y (x, y)

fY (y)

,

when fY (y) > 0. Define

E[X|Y = y] = Z

x

xfX|Y (x|y)dx.

The random variable ψ(Y ) is the conditional expectation of X given Y and denoted as E[X|Y ].