A machine generates rectangular tiles with randomly varying length (L) and width (W). Letthe length follow L ~Gamma(100,10) distribution and the width follow W ~Gamma(50,10)distribution, and define the area of the tile as S = L × W, and its perimeter as Q = 2(L+W).For parts (a-c) below, assume the length and width are independent (i.e. LI W).(a) Show that the expected value of the area is E[S] = 50.(b) Show that the variance of the area is Var[S] = 75.5.(c)Use Chebyshev's inequality to find an upper bound on P(S > 75).For parts (d,e) below, assume the length and width are correlated with Corr(L, W)= -0.5.(d) Show that the expected value of the area is E[S] ~ 49.64645.(e) Find the mean and variance of the perimeter: E[Q] and Var[Q].

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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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