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% binomial_poisson.tex
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\begin{align*}
         S_n &= \sum_{k = 1}^n X_i, \quad k = 0,1,2,\ldots,n\\
\mbox{ \textsf{\textbf{Binomial:}} }   P(S_n = k) &= \dbinom{n}{k} p^k(1 - p)^{n - k}\\
      E(S_n) &= np   \\
      V(S_n) &= np(1 - p) \\
             &   \\
\mbox{ \textsf{\textbf{Poisson:}} }    P(Y = k) &= \frac{\lambda^k}{k!} \exp(- \lambda), \quad y = 0,1,2,\ldots,\\
        E(Y) &= \lambda  \\
        V(Y) &= \lambda  \\
             &    \\
 \lim_{n \rightarrow \infty, p \rightarrow 0}\dbinom{n}{k} p^k(1 - p)^{n - k} &=  \frac {(np)^k}{k!}
 \exp(-np), \quad np > 0
\end{align*}


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