the two questions in the attached file, thank you.

3. Let t E (0, 1) be a fixed constant. Write down the expectations and variances of to for the following cases: (a) X has a Poisson (A) distribution. (b) X has a Binomial (n, p) distribution. (c) X has a geometric (p) distribution, that is, P(X = x) = (1 -p)"p for x = 0, 1,2, .... (d) X has an exp (A) distribution. (e) X has a U[-1, 1] distribution.1. (a) Prove the Markov's inequality, namely P(X | 2 8) 0. [Hint: consider the inequality |X"| 2 1{|X| > }, which is always true, where 1{ } denotes the indicator function.] (b) Let Y1, Y2, ... be independent and identically distributed random variables such that E[Y1] = 0 and E[Yi] is finite for r = 2, 3, 4. Define Yn = n-1 Et_, Yi, for n = 1, 2, .... (i) Using Markov's inequality, show that, for any n = 1, 2, ..., P ( | Y nl > n -1/6) 1, show that lim P(U{Xml > n- 1/6]) = 0. n=k Hence show that P(Un ( 1Xml sn- 1/6 ] ) = 1. k=In=k [Note: The last result implies the famous Strong Law of Large Numbers: P( lim Yn = 0) = 1.]