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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.13. Let X₁, X2,... be i.i.d. r.v.'s with finite expectation and finite variance o², and let X L² →μ as n →∞. n 1/1 X₁. Show that X n i=1 =6. Suppose the moment generating function of X is M(t) = ² et + e²t+e³t. a) Find the mean of X. b) Find the variance of X. c) Find the pmf of X.
- 10. Let T1,T2,….. ,Tk be independent Exponential random variables. Suppose E(T;) 1 i = 1, 2, ... , k. That is, fr,(t) = 2;e¯^i!, t>0, i = 1, 2, ... , k. Denote T, min min (T 1, T2, ... , Tk). %3D а) Show that T min also has an Exponential distribution. What is the mean of T min ?5. Let U U[5, 10] and let X I15.7) (U) be the indicator function of [5, 7) and let Xn I15,7+1/n?)(U). Prove that X, + X with probability one.3. The expectation operator E can be applied to a random vector. Specif- ically, if X = [X1 X2 …… Xn]", then EX = [EX1 EX2 .… EXn]". Show that E[(X – EX)(X – EX)"]= [cov(X¡, X;)]"j=1 •
- 2 Suppose that Z1, Z₂,.. Zn are statistically independent random variables. Define Y as the sum of squares of these random variables: n Y =)Z (n>2) i=1 (a) Express the moment generating function My(t) of the random variable Y in terms of moment generating functions involving the random variables Z², i = 1, ..., n. (b) Determine My(t) for the special case that Z;~ N(0, 1). (c) For the above special case, calculate E[Y] by using the moment generating function.be i.i.d. random variables with expectation 1 and finite 34. Let X1, X, variance o, and set S, = X1+ X2 +-+X,, for n21. Show that VS,-Vn N (0,6*) as and determine the constant 6. and positive, finite4. (a) Consider a nonnegative, integer valued random variable Y , and a random sample X1,., Xn. + Xy) in terms of E(X1) Assume Y and all the X;'s are independent. Find E(X1+ and E(Y). ... (b) Suppose each person who logs onto Amazon.com on Christmas Eve is expected to spend $80. One hundred people are randomly chosen to see how much is spent by them. Each of them visits Amazon.com with probability 60%. How much money do we expect this group of people to spend at Amazon.com?
- 2. Let U1, U2,... be independent random variables, each with continuous distribution that is uniform in the interval [-3, 3]. Let S, = U ++ Un. (a) Compute the moment generating function of U1. (b) Compute the moment generating function of S,. (c) Determine E(S) and E(S) (by any method).7. A drunken man at any time takes a step forward with pr. 1/2, 2 steps backward with pr. 1/4 and stays in place otherwise. He starts at x = 0 at time t = 0 and goes on till time t = T when he passes out. The time T at which he passes out is 1 with probability 1/3, 2 with probability 1/3 and 3 with probability 1/3. (a) Find the exp. of the total number of steps (in any direction) he has taken. (b) Let S be the random variable corresponding to his position when he passes out. Find E[S®J.22. Suppose {Xn, n ≥ 1} are random variables on the probability space (2, B, P) and define the induced random walk by Let So=0, Sn=X₁, n ≥ 1. i=1 T := inf{n > 0: Sn > 0} be the first upgoing ladder time. Prove T is a random variable. Assume we know T (w) < ∞o for all we 2. Prove S, is a random variable.