Question: Suppose \(X\) has an exponential distribution with parameter \(\lambda=1 / 2\). Write down the cumulative distribution function and mean of \(X\). Suppose that \(Y=\sqrt{X}\). (We take the positive square root.) By considering the set \((Y \leq y)\) or otherwise, find the cumulative distribution function of Y. Deduce that the density function \(f_{Y}\) of \(Y\) is \(f_{Y}(y)=0\) for \(y<0\) and \(f_{Y}(y)=y e^{-y^{2} / 2}\) for \(y \geq 0\) Find the mean of \(Y\). Show that the variance of \(Y\) is \(2-\pi / 2\).

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