Question
Let $X$ and $Y^{*}$ be independent random variables, both being equally likely to be any of the values $1,2, \ldots, m$. Show that$$E[|X-Y|]=\frac{(m-1)(m+1)}{3 m}$$
Step 1
Step 1: We start by defining the expectation of the absolute difference between $X$ and $Y^{*}$ as follows: $$ E[|X-Y^{*}|] = \sum_{i=1}^{m} \sum_{j=1}^{m} |i-j| \cdot P(X=i, Y^{*}=j) $$ Since $X$ and $Y^{*}$ are independent and equally likely to be any of the Show more…
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