4. Let 1A be the indicator of event A. That is, the random variable 14 takes the value 1 if A occurs and is 0 if AC occurs. (a) What is the support of 1A? (b) What is the p.m.f. of 1A? (c) Compute E[1A].

Suppose X is a random variable with E(X) = 0 and Var(X) = 2. What is E(X2 )? a). 1 b). 2 c). 3 d). 4 e). none of above

1. The random variables §, §1, §2,... are independent and identically distributed with 1/4 and P(§ = j) = c/j for j = 1,2,3. Let X₁ = 0 and Xn = max(§₁, ..., Èn) for n = 1, 2,.... distribution P(§ = 0) = (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix. (d) Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). (e) Calculate P(Xn = 0). (f) Calculate P(X₁ = 3, X₂ = 1|X₁ = 3).

8) Consider the following Random Variable X X = k Pr(X=k) 3 .2 7 .1 1 .4 .2 .1 Find the following: a) Pr(X > 4) b) E(X) c) V(X)