E2 202 (Aug–Dec 2015)Homework Assignment 4

Discussion: Friday, Sept. 18

1. For this problem, to avoid technical complications, assume that all the expected values needed for the defini-tions are finite. Let X be a discrete random variable with pmf pX . Set X , {x : pX(x) > 0}. Using therelevant definitions, verify the following:

(a) For any subset A ⊆ X , we have

E[X|A] =∑

x∈A xpX(x)∑x∈A pX(x)

.

(b) Let Y be a discrete r.v. jointly distributed with X , and let ψ(X) = E[Y |X]. Then, for any (Borelmeasurable) function g : R→ R, we have

E[ψ(X)g(X)] = E[Y g(X)].

2. Let X and Y be jointly continuous random variables with a joint density fXY , and let ψ(X) = E[Y |X].Verify that for any (Borel-measurable) function g : R→ R, we have

E[ψ(X)g(X)] = E[Y g(X)].

Again, assume that all the relevant expected values are finite.

3. Let X ∼ EXP(λ) be an exponential random variable with parameter λ, i.e., FX(x) = 1 − e−λx for x > 0,and FX(x) = 0 otherwise. Evaluate E[X] and E[X|X > t].

4. Let (X,Y ) be uniformly distributed over the triangular region with vertices (0, 0), (2, 0) and (2, 1).

(a) Determine the marginal densities of X and Y ; also determine E[X] and E[Y ].

(b) Find the conditional density fY |X(y|x). Be sure to specify the values of x for which this conditionaldensity is defined.

(c) Find the conditional expectation E[Y |X = x]. Be sure to specify the values of x for which this condi-tional expectation is defined.

(d) Determine the distribution function of the random variable E[Y |X], and also determine its density, if itexists.

(e) Explicitly compute E[E[Y |X]

]and verify that it equals E[Y ].