Chapter 3: Problem 23
Let \(X\) and \(Y\) be independent random variables, each with a distribution that is \(N(0,1) .\) Let \(Z=X+Y .\) Find the integral that represents the cdf \(G(z)=\) \(P(X+Y \leq z)\) of \(Z .\) Determine the pdf of \(Z\). Hint: We have that \(G(z)=\int_{-\infty}^{\infty} H(x, z) d x\), where $$ H(x, z)=\int_{-\infty}^{z-x} \frac{1}{2 \pi} \exp \left[-\left(x^{2}+y^{2}\right) / 2\right] d y $$ Find \(G^{\prime}(z)\) by evaluating \(\int_{-\infty}^{\infty}[\partial H(x, z) / \partial z] d x\).
Short Answer
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Key Concepts
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