题目020 · 概率
No Fixed Points Among the First Three
How many permutations of {1,2,...,8} have no fixed point among positions 1, 2, and 3?
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中文题目题目215 · 概率
Distribution of Dice Sums via Probability Generating Functions
Let $X_1, X_2, \ldots, X_n$ be iid rolls of a fair $d$-sided die, so each $X_i$ is uniform on $\{1, 2, \ldots, d\}$. Let $S_n = X_1 + X_2 + \cdots + X_n$. (a) Derive the PGF $G_{X_1}(s) = E[s^{X_1}]$ in closed form. (b) Write the PGF of $S_n$ and use it to derive $E[S_n]$ and $
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