Let \( X \) and \( Y \) be discrete random variables with joint PMF \( P_{X, Y}(x, y) \) that is zero except when \( x \) and \( y \) are integers. Let \( W=X+Y \) and show that the PMF of \( W \) satisfies \[ P_{W}(w)=\sum_{x=-\infty}^{\infty} P_{X, Y}(x, w-x) \]
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Step 1
We need to prove that \( P_W(w) \), the PMF of \( W \), is equal to the sum of the joint PMFs of \( X \) and \( Y \) over all possible values of \( X \), where \( Y \) is adjusted accordingly to maintain the sum \( W = X + Y \). Show more…
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