Let X1, X2, · · be i.i.d. random variables ~ X, and let N be random variable taking values in {0, 1, 2, ·}. (a) Show that E[X1+X2+ • ··+XN] = (EN)(EX) if N is indepen- dent of X1, X2, .... (b) Is it true in general that E[X1+X2 + · · + XN] = (EN)(EX)?

Q: 4. (a) Consider a nonnegative, integer valued random variable Y, and a random sample X1, ·.. , Xn.…

A: Expectation gives the most likely value of a random variable.

Q: We’ll say that a permutation T = (T(1),.….., T(n)) contains a swap if there exist i, je{1,..,n} so…

A: a) Let we take n > 1. Let Xi be a random variable if i is a partner in a swap or not. The values…

Q: 3. If X and Y are independent uniform (0, 1) random variables, show that 2 for a > 0 (a + 1)(a + 2)…

A: X and Y are independent U(0,1) random variables:Objective: Show that:

Q: 5. (a) Suppose X is a discrete random variable with range in the positive integers {1, 2, 3, 4, . .…

A: Given that X is a discrete random variable with range in the positive integers {1, 2, 3, 4, . . .}.

Q: Suppose X and Y are two independent and identically distributed geometric random rariables. The pmd…

A: PMF of X is given by, P(X=x) = p(1-p)x-1 , x=1,2,3.... Now,  ⇒P(X≥x)= ∑t=x∞ P(X=t)= ∑t=x∞p(1-p)t-1 =…

Q: Let M and N be two independent events such that P(M) = 1/3 and P(N) = 1/4. Find P(M/N')

A:

Q: Suppose that E and F are two events such that Pr[E] = 17/29, Pr[F] = 13/29, and Pr[EN F] = 8/29.…

A: Given, P(E) = 1729P(F) = 1329P(E∩F)=829

Q: (1) E(E(X)) = E(X) (2) E(X – E(X)) = 0 (3) Var(X) = E(X²) – [E(X)]² (4) Given constants a and b,…

A: Discrete random variable is defined as a variable which can take only specific values. For example…

Q: Let X1 and X2 be independent random variables with respective binomial distributions b(3,1/2) and…

A: The provided information is as follows:The  and  are two independent variables.The  follows binomial…

Q: 2. Let U1, U2,... be independent random variables, each with continuous distribution that is uniform…

A: The problem can be solved using the concept of expectation and moment generating function.

Q: 3- Show that, if the random variables X and Y are N(0 6,0",r) , thên 1 a) E{ fy(y|x)} = =exp{- 20…

A:

Q: 7. A discrete random variable X taking non-negative integer values has probability gen- erating…

A:

Q: A simple example of a random variable is the indicator of an event A, which is denoted by IA: IA(W)…

A: Indicator function is defined as   IA(ω) =1 if ω ∈ A, 0 otherwise

Q: (a) Var(X + Y ) = Var(X)+ Var(Y) (b) Ε(XY) - (EX) (EYΥ) (c) E(X + Y ) = EX+ EY

A: Since we answer up to three subparts of a problem, we'll be answering the first three sub parts…