15.3.1 The Poisson distribution

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  $$\begin{eqnarray*}p(k;\lambda) = e^{-\lambda} \frac{\lambda^k}{k!} \end{eqnarray*}$$
  
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  $$\begin{eqnarray*}\lim_{n \to \infty, p \to 0} b(k;n,p) = p(k;\lambda)\ \ \ (\lambda = np) \end{eqnarray*}$$
  
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  $$\begin{eqnarray*}b(k;n,\frac{\lambda}{n}) &=& \frac{n(n-1)\cdots(n-(k-1))}{k!} ... ...bda}}\Bigr\}^{-\lambda} \Bigl(1-\frac{\lambda}{n}\Bigr)^{k} \\ \end{eqnarray*}$$
  
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  $$\begin{eqnarray*}\lim_{n \to \infty} b(k;n,\frac{\lambda}{n}) = e^{-\lambda} \frac{\lambda^k}{k!} \end{eqnarray*}$$
  
  
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  $$\begin{eqnarray*}P_i(k) = p(k;\lambda_i) = e^{-\lambda_i} \frac{\lambda_i^k}{k!} \end{eqnarray*}$$
  
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  $$\begin{eqnarray*}b(k;n,\frac{\lambda}{n}) = { }_nC_k \Bigl(\frac{\lambda}{n}\Bigr)^k \Bigl(1-\frac{\lambda}{n}\Bigr)^{n-k} \end{eqnarray*}$$

  
  

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  $$\begin{eqnarray*}p(k;\lambda) = e^{-\lambda_i} \frac{\lambda_i^k}{k!} \end{eqnarray*}$$

  
  

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