It's time to roll the dice.
\(\mathtt{Definition}\)
令 \(X\) 为取值非负的随机变量，那么 \(X\) 的概率生成函数 \(\mathtt{Probability\ Generating\ Function}\) 为
\[\begin{aligned}
G_x(z) = \sum_{k \ge 0} \mathrm{Pr}(X = k) z^k
\end{aligned}
\]
根据上式可以得知该生成函数各项系数均非负，且其和为 \(1\) ，即是
\[\begin{aligned}
G_x(1) = 1
\end{aligned}
\]
那么反过来，任何非负系数且满足 \(G_x(1) = 1\) 的幂级数 \(G_x\) 均为某随机变量的生成函数。
\(\mathtt{Average \ \& \ Variance}\)
\(\mathtt{Average}\)
\[\begin{aligned}
\mathrm{E}(X) &= \sum_{k \ge 0} k \mathrm{Pr}(X = k) \\
&= \sum_{k \ge 0} \mathrm{Pr}(X = k) k \cdot 1^{k - 1} \\
&= G_{x}^{'} (1)
\end{aligned}
\]
进而有
\[\begin{aligned}
\mathrm{E}(X^{\underline{k} }) = G_x^{k}(1) (k \not = 0)
\end{aligned}
\]
\(\mathtt{Variance}\)
\[\begin{aligned}
\operatorname{D}(X)&=\operatorname{E}(X^2)-\operatorname{E}(X)^2\\
&=G_{X}^{''}(1)+G_{X}^{'}(1)-G_{X}^{'}(1)^2
\end{aligned}
\]
即是说在知道 \(G_{X}^{''}(1)\) 和 \(G_x^{'}(1)\) 的情况下就可以得到 \(\mathrm{D}(X)\) 。
\(\mathtt{Multiplication}\)
若两个随机变量 \(X, Y\) 互相独立，那么有
\[\begin{aligned}
\operatorname{Pr}(X+Y=n)=&\sum_{k}\operatorname{Pr}(X=k \wedge Y=n-k)\\
=&\sum_{k}\operatorname{Pr}(X=k)\operatorname{Pr}(Y=n-k)\\
\end{aligned}
\]
写成卷积的形式
\[\begin{aligned}
G_{X+Y}(z)=G_{X}(z)G_{Y}(z)
\end{aligned}
\]
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