다항분포

\begin{bmatrix} Y_{1} \\ Y_{2} \\ \end{bmatrix} \sim multinominal (n, \begin{bmatrix} Y_{1} \\ Y_{2} \\ \end{bmatrix} )

P(Y_{1}=y_{1}, Y_{2}=y_{2}) = \dfrac {n!} {y_{1}!y_{2}!(n-y_{1}-y_{2})!} \, p_{1}^{y_{1}} \, p_{2}^{y_{2}} \, (1-p_{1}-p_{2})^{n-y_{1}-y_{2}}

P\,(Y_{1} = y_{1}) = \sum_{y_{2}=0}^{n-y_{1}} \,p\,(Y_{1} = y_{1}, \,Y_{2} = y_{2}) \\[20pt] \begin{align*} &= \sum_{y_{2}=0}^{n-y_{1}}~\cdot~ \dfrac {n!} {y_{1}!~y_{2}!~(n-y_{1}-y_{2})!}~ p_{1}^{y_{1}}~ p_{2}^{y_{2}}~ (1-p_{1}-p_{2})^{n-y_{1}-y_{2}} \\[20pt] &= \frac{n!}{y_{1}!}~\cdot~ p_{1}^{y_{1}}~ \sum_{y_{2}=0}^{n-y_{1}} \dfrac {n!} {y_{2}!~(n-y_{1}-y_{2})!}~ p_{2}^{y_{2}}~ (1-p_{1}-p_{2})^{n-y_{1}-y_{2}} \\[20pt] &= \binom{n}{y_{1}}~ p_{1}^{y_{1}}~ \{p_{2} + (1-p_{2}-p_{2})\} ^{n-y_{1}} \\[20pt] &= \binom{n}{y_{1}} ~p_{1}^{y_{1}}~ (1-p_{1})^{n-y_{1}} \end{align*}