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5.4-9. Let X and Y, with respective pmfs f(x) and g(y), be independent discrete ran- dom variables, each of whose support is a subset of the nonnegative integers 0, 1,2, Show that the pmf of W - X + Y is given by the convolution formula x=0 Hint: Argue that h(w) P(Ww) is the probability of the w 1 mutually exclusive events (x, y-w-x), 0,,..., w.

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Answer #1

Given that x and y are independent.

Hence f(x,y) = f(x) g(y)

W = x+y This implies that x =x and y = w-x

Hence x = x and y = w-x, for x = 0,1,2......w

Hence P(w) = P(x =x, y = w-x) = f(x) g(w-x), x = 0,1,2,....w

h(w)=\sum_{x=0}^{w}f(x)g(w-x), w=0, 1, 2, ...

Hence proved

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