Consider two fair $k$-sided dice with the numbers 1 through $k$ on their faces, obtaining values $X_1$ and $X_2$. What is $\mathbb{E}[\max(X_1, X_2)]$ and what is $\mathbb{E}[\min(X_1, X_2)]$.
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HINT: $\min(x_1,x_2) + \max(x_1,x_2) = x_1 + x_2$, so it is enough to evaluate either expected value. Also $\max(x_1,x_2) - \min(x_1,x_2) = \vert x_2-x_1\vert$. Therefore, finding $\mathbb{E}(\vert X_2-X1 \vert)$ allows to determine expectations needed: $$ \begin{eqnarray} \mathbb{E}\left( \vert X_2- X_1\vert \right) &=& \sum_{n_1=1}^k \sum_{n_2=1}^k \mathbb{P}(X_1=n_1) \mathbb{P}(X_2=n_2) \vert n_2 - n_1 \vert \\ &=& 2 \sum_{n_2=1}^k \sum_{n_1=1}^{n_2-1} \mathbb{P}(X_1=n_1) \mathbb{P}(X_2=n_2) (n_2 - n_1 ) \\ &=& 2 \sum_{n_2=1}^k \sum_{n_1=1}^{n_2-1} \mathbb{P}(X_1=n_1) \mathbb{P}(X_2=n_2-n_1) n_1 \end{eqnarray} $$ Can you finish this ? |
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