Question: The random variable \(Y\) has the standard normal density with mean 0 and variance 1 , \(Y \sim \mathcal{N}(0,1)\). Find the distribution and density functions of \(V=Y^{2}\)

The moment generating function \(M_{X}(t)\) of a random variable \(X\) is defined by \(M_{X}(t)=\) \(E\left[e^{t X}\right] .\) If \(X \sim \mathcal{N}(0,1)\), show that \(M_{X}(t)=e^{t^{2} / 2}\)

Let \(X\) and \(Y\) be independent standard normal random variables, and \(Z=X Y\). Find \(M_{Z}(t)\), either by direct calculation or via the tower law in the form \(E\left[e^{t Z}\right]=\) \(E\left[E\left[e^{t Z} \mid Y\right]\right]\)

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