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(Solved): 3. [Distribution of a function of an exponential random variable. ] \( X \) is an exponential rand ...
![3. [Distribution of a function of an exponential random variable. ] \( X \) is an exponential random variable with mean 1. Le](https://media.cheggcdn.com/media/e8c/e8cea276-65e0-4ab8-b073-1b69876b3cc5/php2vafYe)
3. [Distribution of a function of an exponential random variable. ] \( X \) is an exponential random variable with mean 1. Let \( Y=(1+X)^{2} \). (a) Find the CDF of \( Y \). (b) Find the pdf of \( Y \). (c) Calculate \( E[Y] \). 4. [Absolute value of a random variable] Express the pdf of \( |Y| \) in terms of the pdf of \( Y \).
Expert Answer
(a) Fy(y)=Fx(u?1(y)) Y=(1+X)2 u?1(y)=X=±Y2?1 For any exponential distribution cdf = 1?e??x Fy(y)=1?e??(±Y2?1) Fy(y)=1?e??(+Y2?1),Y?1 ,for CDF to be fi