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Question 4 (a) The continuous random variables X and Y are distributed uniformly on (0, 1) and exponentially with mean 1, respectively. X and Y are independent. (i) Use the convolution formula to show that the random variable Z = X +Y has probability density function (pdf) 0, 1-e-*, (e- 1) e-, z>1. fz(z) = 0

Question 4 (a) The continuous random variables X and Y are distributed uniformly on (0, 1) and exponentially with mean 1, respectively. X and Y are independent. (i) Use the convolution formula to show that the random variable Z = X +Y has probability density function (pdf) 0, 1-e-*, (e- 1) e-, z>1. fz(z) = 0

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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please send solution for part a iii
Question 4 (a) The continuous random variables X and Y are distributed uniformly on (0, 1) and exponentially with mean 1, respectively. X and Y are independent. (i) Use the convolution formula to show that the random variable Z = X +Y has probability density function (pdf) = (2)25 (e - 1) e-, z> 1. 1-e=², 0 <z<%; (ii) Show that fz(z) is a valid pdf. (iii) Let Fz(z) be the cumulative distribution function (cdf) of Z. Show that z<0; 0, z+e-1, Fz(2) 0<z<1; %3D e(1- e) +1, z> 1. and hence find the median of Z. (b) The random variable X has pdf S 6x (1-2), 0 < x < 1; otherwise. fx (x) = Determine the pdf of Y = X/(1 X). (c) The random variable X has pdf
Transcribed Image Text:Question 4 (a) The continuous random variables X and Y are distributed uniformly on (0, 1) and exponentially with mean 1, respectively. X and Y are independent. (i) Use the convolution formula to show that the random variable Z = X +Y has probability density function (pdf) = (2)25 (e - 1) e-, z> 1. 1-e=², 0 <z<%; (ii) Show that fz(z) is a valid pdf. (iii) Let Fz(z) be the cumulative distribution function (cdf) of Z. Show that z<0; 0, z+e-1, Fz(2) 0<z<1; %3D e(1- e) +1, z> 1. and hence find the median of Z. (b) The random variable X has pdf S 6x (1-2), 0 < x < 1; otherwise. fx (x) = Determine the pdf of Y = X/(1 X). (c) The random variable X has pdf
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