5. Suppose X is a random variable. (a) X is independent of itself if and only if there is some constant c such that P[X = c] = 1. (b) If there exists a measurable g: (R, B(R)) → (R, B(R)), such that X and g(X) are independent, then prove there exists c E R such that P[g(X) = c] = 1.
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![5. Suppose X is a random variable.
(a) X is independent of itself if and only if there is some constant c such that
P[X = c] = 1.
(b) If there exists a measurable
g: (R, B(R)) → (R, B(R)),
such that X and g(X) are independent, then prove there exists c ER such
that
P[g(X) = c] = 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65db2aa6-eb47-44d1-842d-3deeee985571%2F077c48aa-387d-4f5a-a0da-b94585ab091b%2Fhicghov_processed.jpeg&w=3840&q=75)
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- 4. (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?Let X and Y be independent uniform random variables on (0,1). Let Z = [1/(X+Y)]. (Recall that is the largest integer ≤ x.) (a) Find P(Z = 0). (b) Find E[Z].Prove that for a continuous random variable X,E (aX+ b) = aE (X) + b.
- = B1. Let X₁ and Y; be random variables with Var(X;) = o² and Var(Y₂) o for all ie {1,...,n}. Assume that each pair (X₁, Y;) has correlation Corr(X, Y) = p, but that (X₁, Y₂) and (X₁, Y;) are independent for all i j. (a) What is Cov(X₁, Yi) in terms of Ox, Oy (b) Show that Cov(X₁,Y) = poxoy/n, where Y is the average of the Y₁. (c) Determine Cov(X, Y). and p?Let u(X) be a nonnegative function of a random variable X. It can be shown that P(u(X) >= c) <= E(u(X)) / c Suppose X is a random variable with momonet generating function Mx(t). Show that P(X>= a) <= exp(-ta) * Mx(t) Thank youSuppose that X and Y are random variables with E(XY) = E(X)E(Y). Then X and Y * independent. Also Var(X + Y) = Var(X) + Var(Y) true.
- Suppose X and Y are two independent and identically distributed geometric random variables. The pmd of X is P(X = x) = p(1 – p)*-1 for x = 1,2, ... Show that P(X >x) = (1- p)-1. Show that P(X 2r+T)|(X > T)] = P(X > x) where T is a positive integer, i.e., Find the moment-generating function of X. Let Z = X +Y. What is the moment generating function of Z?Let X1, X2, X3 be random variables such that Var(X1) = 5, Var(X2) = 4, Var(X3) = 7, cov(X1, X2) = 3, cov(X1, X3) = -2 and X2 and X3 are independent. Find the covariance between Y1 = X1 – 2X2 + 3X3 and Y2 = -2X1 + 3X2 + 4X3. %3DLet Y₁ = (x,-X₂), where X, and X₂ are independent random variables and each of them X, 2 is distributed as x² (2). Find the p.d.f. of Y₁.
- 7. Suppose random variables X and Y are independent. Let g(x) and h(y) be any bounded measurable functions. Show that Cov(g(X), h(Y)) = 0.(b) Let X and Y be independent random variables. Prove that f(X) and g(Y) are independent random variables for any Borel functions f and g on R.b) A continuous random variable X has the p.d.f f(x) = {A(2 – x)(2 + x), 0 < x < 2, Find (i) the value of A, (ii)P(X <1)(iii) P(1 < X <2). l0, otherwis ------------