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Exercise 5.2.4

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Suppose $Z = X + Y$ , we compute the probability $P (Z \leq z) = P (X + Y \leq z) = \int_{x + y \leq z} p_{x} (x) p_{y} (y) dxdy = \int_{x} \int^{z - x} p_{x} (x) p_{y} (y) dxdy$ , take derivative w.r.t. $z$ we have the probability density function $p (z) = \int_{x} p_{x} (x) p_{y} (z - x) dx$ , which is the convolution between $p_{x}$ and $p_{y}$ .

niuers

2020-03-20 00:00

Exercise 5.2.4

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