please give detail solution. Let X be an r.v. with uniform distribution on [0, 1]. Show that X 2 ∼ Beta(1,1).
please give detail solution. Let X be an r.v. with uniform distribution on [0, 1]. Show...
U means Uniform distribution
2. Let X be a r.v. distributed as U(α, β). Show that its ch. f. and m.g.f.x and Mr, respectively, are given by and IM x it(β-a) , t(B-a) ii) By differentiating (ax, show that E(X)-(α + β) / 2 and T 2 (X)-(α-β)2 / 12.
Let X be a Gaussian r.v. with mean 5 and sigma 10. Let Y be an independent exponential r.v. with lambda 3. Let Z be an independent continuous uniform r.v. in the interval [-1,1]. a. (5) Compute E[X+Y+Z]. b. (5) Compute VAR[X+Y+Z].
Exercise 4.8: Suppose that X1, X2,..., Xn is a random sample of observations on a r.v. X, which takes values only in the range (0, 1). Under the null hypothesis Ho, the distribution of X is uniform on (0, 1), whereas under an alternative hypothesis, њ, the distribution is the truncated exponential with p.d.f. 0e8 where 6 is unknown. Show that there is a UMP test of Ho vs Hi and find, roximately, the critical region for such a test...
True or False: Let X be an r.v. with mean up = 0. The transformed variable Y = X also has a mean uy = 0. Let X be an r.v. with o z. The transformed variable Y = X2 has oy = 02. Let X be an r.v. defined over -1 < x < 1. The transformed variable Y = X4 - X has exactly 3 terms in its PDF,
Let X be a R.V. with a gamma distribution and the following parameters (X~(α, 1)). What is the pdf and the cdf of Y = X/β, where β > 0 . What is the name of this type of distribution?
Let X be an uniform distribution between 0 and 1, Y be an uniform distribution between -5 and 3, and they are independent. Calculate the pdf, expectation, and the variance of the followings 1. 4X
Let X be a r.v. with probability density function f(x)-e(4-x2), -2 < otherwise (a) What is the value of c? (b) What is the cumulative distribution function of X? (c) What is EX) and VarX
Let pdf of a r.v. X be given by f(x) = 1, 0<x< 1. Find Elet).
5. Let X have the uniform distribution U(0, 1), and let the conditional distribution of Y, given X = x, be U(0, x). Find P(X + Y ≥ 1).
Let X and Y be independent exponential(1) RVs (f(x) e 10). Show that uniform(0, 1) distribution. Hint: consider defining the auxiliary X/(X Y) has a RV XY [12