7. Suppose that Xi,..., Xk are independent random variables, and X, ~ Exp(B) for i =...
suppose Xi, X2, . . . , X, are i.id. random variables with Xi ~ exp(A). Show that Σ-x, ~ T(n, t).
Suppose X = Exp(1) and Y= -ln(x) (a)Find the cumulative distribution function of Y . (b) Find the probability density function of Y . (c) Let X1, X2, ... , Xk be i.i.d. Exp(1), and let Mk = max{X1,..... , Xk)(Maximum of X1, ..., Xk). Find the probability density function of Mk.(Hint: P(min(X1, X2, X3) > k) = P(X1 >= k, X2 >= k, X3 >= kq, how about max ?) (d) Show that as k → 00, the CDF...
Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random variables with mhean μ and variance a) Compute the expected value of W b) For what value of a is the variance of W a minimum? σ: Let W-aX + (1-a) Y, where 0 < a < 1. Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random...
(3) Consider a sequence of independent and identically distributed random variables such that Xk-0, with common mean EĮXk] = 1. Define the Xi, X2, ,Xp, sequence k=1 (a) Compute E[ (b) Show that (3) Consider a sequence of independent and identically distributed random variables such that Xk-0, with common mean EĮXk] = 1. Define the Xi, X2, ,Xp, sequence k=1 (a) Compute E[ (b) Show that
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
6. (10 points) Suppose X – Exp(1) and Y = -In(X) (a) Find the cumulative distribution function of Y. (b) Find the probability density function of Y. (c) Let X1, X2,...,be i.i.d. Exp(1), and let Mk = max(X1,..., Xk) (Maximum of X1, ..., Xk). Find the probability density function of Mk (Hint: P(min(X1, X2, X3) > k) = P(X1 > k, X2 > k, X3 > k), how about max ?) (d) Show that as k- , the CDF of...
2. Let Xi exp(1) and X2 ~ variables with rate 1. Let: erp(1) be independent and identically-distributed exponential random (a) What is the cdf of X1? b) What is the joint pdf of (Xi, X2)? (c) What is the joint pdf of (Y, Z)? d) What is the marginal pdf of z?
4. Lct Xi, i 1,2,3, be three independent random variables and let Y -XiX2+X3 Find the pdf of Y and identify the distribution of Y when X, have the following distributions. Show your work. (b) X V i. The density for a chi-square random variable matches that of a gamma random variable with α = vi/2 and β = 2.
Let Xi Pn(2) and X2 Pn(5) be two independent random variables and it that y = Xi + X-Pn(7). is shown (a) Given Y-n, n 20, what are the possible values of X1? (b) Calculate the conditional distribution of Xi given Y-n for n 2 0. Let Xi Pn(2) and X2 Pn(5) be two independent random variables and it that y = Xi + X-Pn(7). is shown (a) Given Y-n, n 20, what are the possible values of X1? (b)...
in 4. Suppose that {Xk, k > 1} is a sequence of i.i.d. random variables with P(X1 = +1) = 1. Let Sn = 2h=1 Xk (i.e. Sn, n > 1 is a symmetric simple random walk with steps Xk, k > 1). (a) Compute E[S+1|X1, ... , Xn] for n > 1. Hint: Check out Example 3.8 in the lecture notes (Version Mar/04/2020) for inspiration. (b) Find deterministic coefficients an, bn, Cn possibly depending on n so that Mn...