## Due Monday,  April 20

### 2.1(10points) If {Xn}is a sequence of independent and identically distributed v.’s not constant a.e., then P[Xnconverges] = 0.

2.2(10points) Suppose limn+ P[|XnX| > s] = 0 for any s > 0 and P[X x] = 0. Show that P[{X x}O{Xn x}] 0.

2.3(10 points) Let α be completely normal. Show that by looking at theexpansion of α in some scale we can rediscover the complete works of Shakespeare from end to end without a single misprint or interruption.

### 2.4(10points) For any sequence of v.’s {Xn}, (a) Xn→0 a.e. would result in Sn/n → 0 a.e. (b) Xn→ 0 in Lp would result in Sn/n → 0 in Lp for p ≥ 1.  高级概率作业代写

2.5(10points) Let {Xn, n  1} be a sequence of independent, identically distributed r.v.’s; also, let τ be a positive integer-valued v. that is independent of the Xn’s.

Suppose that both τ and X1 have finite second moments, then

σ2(Sτ ) = E[τ ]σ2(X1) + σ2(τ )(E[X1])2.

2.6(10points) Let {Xn, n  1} be a sequence of independent, identically distributed r.v.’s;  also for some finite l, we  have Σl pk = 1 where each pk P[X1 = k]. Let

Nk(n) be the number of values of j = 1, 2, …, n such that Xj = k. Show that

in addition, find the limit.

### 2.7(10points) Suppose that supnfdµn<+∞for a nonnegative function fsuch that f (x) → +∞ as x → ±∞. Show that {µn}is tight.  高级概率作业代写

2.8(10 points) Let f be the ch.f. of the p.m. µ. For each x0, showthat

2.9(10points) Show that the f. for the standard normal Z is f (t) = et2/2.  高级概率作业代写

2.10(10 points) For a Poisson variable Yλsuch that P[Yλ = n] = eλλn/(n!) for n = 0, 1, …, show that (Yλ λ)/λ Z the standard Normal as λ +.