기하 ⇒ 음이항

\begin{align*} \text{ex. }~&X \sim Geometric.~~i.i.d \\[5pt] &Y \sim X_1 + X_2 \end{align*}

P(Y_2 = y) = P(X_1 + X_2 = y) \\[20pt] \begin{align*} &= \sum_{s=1}^{y} P(X_1 = s, X_2 = y-s) \\[20pt] &= \sum_{s=1}^{y}~ P(X_1 = s)~P(X_2 = y-s) \\[20pt] &= \sum_{s=1}^{y}~ pq^{s-1}~I\big(s \in \{1,~2,~\cdots\}\big)~ pq^{y-s}~I\big(y-s \in \{1,~2,~\cdots\}\big) \\[20pt] \end{align*}

\begin{align*} \sum_{s=1}^{y-1} ~pq^{s-1}~pq^{y-s-1} =&~~p^2~\sum_{s=1}^{y-1}~q^{y-2} \\[10pt] =&~~(y-1)~p^2~q^{y-2} \end{align*}

\begin{align*} &X_i \sim geo(p) \\[10pt] &M_{X_i}(t) = \sum_{x=1}^{\infty} \cdot~e^{tx}~P(X_i=x) \end{align*}