(5 pts) Let U be a random variable following a uniform distribution on the interval [0,1...
(5 pts) Let U be a random variable following a uniform distribution on the interval [0, 1]. Let X=2U + 1 Calculate analytically the variance of X. (HINT : Elg(z)- g(z)f(x)dr, and the pdf. 0 < z < 1 0 o.t.w. f(x) of a uniform distribution is f(x) =
Please show your works (3) (5 pts) Let U be a random variable following a uniform distribution on the interval [0, 1 Let X-2U1 Calculate analytically the variance of X. (HINT: E[J(x)-「 f(x) of a uniform distribution is f(x) g(x)f(x)dz, and the pdf. 0 o.t.w.
(20 pts) Let U be a random variable following a uniform distribution on the interval [0 Let X=2U + 1 (a) Is X a random variable? Why or why not? (b) Calculate E[X] analytically
Please show your works (2) (20 pts) Let U be a random variable following a uniform distribution on the interval [0,1 Let (a) Is X a random variable? Why or why not? (b) Calculate EX] analytically
Other answers show 12 in the denominator when solving for the final answer, please explain why. (3) (5 pts) Let U be a random variable following a uniform distribution on the interval [0, 1]. Let X-2U1 Calculate analytically the variance of X. (HINT: El( f(x) of a uniform distribution is f(x) g) f(x)dx, and the p.d.f 0 о.t.u.
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a) Show that V 1 - U is also a uniformly distributed random variable on the interval (0,1) (b) Show that X-In(U) is an exponential random variable and find its associated parameter (c) Let W be another random variable that is uformly distributed on (0,1). Assume that U and W are independent. Show that a probability density function of Y-U+W is y, if y E...
Suppose that U is a random variable with a uniform distribution on (0,1). Now suppose that f is the PDF of some continuous random variable of interest, that F is the corresponding CDF, and assume that F is invertible (so that the function F-1 exists and gives a unique value). Show that the random variable X = F-1(U) has PDF f(x)—that is, that X has the desired PDF. Hint: use results on transformations of random variables. This cute result allows...
U is Uniform distribution here Let X ~ U[0,1] and Y = max {,x) (a) Is Y a continuous random variable? Justify (b) Compute E[Y]. (Hint: Note that when a (Hint: Note that when a-, max 1.a- , and when a > ļ, max | , a- ax {3a, and when a > a
Let the random variable X have a uniform distribution on [0,1] and the random variable Y (independent of X) have a uniform distribution on [0,2]. Find P[XY<1].