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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)
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...
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5. 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 ={ [ 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 (d) Suppose that Compute the mean of V. Optional. Obtain g. rb d # [ext, r)dr = [(tr)dr. (t,r)dr dt a a f(u) = 6u(1 — u), 0 < u < 1.
Transcribed Image Text:5. 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 ={ [ 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 (d) Suppose that Compute the mean of V. Optional. Obtain g. rb d # [ext, r)dr = [(tr)dr. (t,r)dr dt a a f(u) = 6u(1 — u), 0 < u < 1.
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