Let U and R be independent continuous random variables taking values in [0, 1]. We assume that R is uniform and that U has PDF f, CDF F and mean . Define the random variable V sa follows V = { R+ (1 - R)U if R≤ 1/2 RU if R > 1/2 (a) Obtain the mean of V in terms of u. (b) Obtain an integral expression of the CDF G of V in terms of F. (c) Deduce an integral expression of the PDF g of V in terms of f. Hint: You may assume that in this case

Let U and R be independent continuous random variables taking values in [0, 1]. We assume that R is uniform and that U has PDF f, CDF F and mean . Define the random variable V sa follows V = { R+ (1 - R)U if R≤ 1/2 RU if R > 1/2 (a) Obtain the mean of V in terms of u. (b) Obtain an integral expression of the CDF G of V in terms of F. (c) Deduce an integral expression of the PDF g of V in terms of f. Hint: You may assume that in this case

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

Let Z₁ and Z₂ be independent standard normal random variables. Let pe [-1, 1]. Find a matrix L such that has a distribution. X = LZ N (1). (1))
Let JO, J1,..., J4 independent random variables according to the Ber (r;) law, where i = 0, 1,..., 4, respectively. We define the random variables Xi = min {JO + Ji, 1}, for i = 1, 2, 3, (a) Find the law of Xi , for each i = 1, 2, 3, 4. (b) Find the law of (X1, X2, X3, X4).
Let X1, X2, · · · , Xn be uniform i.i.d. random variables between [0, 1].Find the CDF and PDF of X = min(X1, X2, · · · , Xn)
Let f(x, y) = x + y for 0 < x < 1 and 0 < y < 1 The Conditional Variance of Y when X = ; is
!
Example: Suppose that X u = (120, 80) and covariance matrix (X1, X2) has a bivariate Normal distribution with mean 6. 3 Σ= 5 What are the mean and variance of a' X, where a = (1, –1)T?
Let X and Y have joint PDFfX,Y (x, y) = 3x^2y + 3xy^2 if 0 < x < 1, and 0 < y < 1 0 otherwise.a) Find the covariance of X and Y .b) Find Var(X).c) Find the correlation of X and Y (X and Y have the same variance.)