mean_and_variance_of_poisson_distribution
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| mean_and_variance_of_poisson_distribution [2020/11/21 02:15] – [Variance] hkimscil | mean_and_variance_of_poisson_distribution [2024/10/28 07:51] (current) – hkimscil | ||
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| + | ====== Mean and Variance of Poisson Distribution ====== | ||
| ====== Mean ====== | ====== Mean ====== | ||
| Mean Poisson distribution = $\lambda$ | Mean Poisson distribution = $\lambda$ | ||
| Line 11: | Line 12: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | 우선 Taylor series을 이용하면 | + | 우선 |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| e^{a} = \sum_{y=0}^{\infty} \frac{a^y}{y!} \\ | e^{a} = \sum_{y=0}^{\infty} \frac{a^y}{y!} \\ | ||
| Line 81: | Line 82: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | 따라서 | ||
| + | \begin{eqnarray*} | ||
| + | E[X(X-1)] & = & E[X^2-X] = E(X^2) - E(X) \\ | ||
| + | & = & \lambda^2 | ||
| + | E(X^2) & = & \lambda^2 + \lambda \\ | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | 다시 원래대로 돌아가서 | ||
| + | \begin{eqnarray*} | ||
| + | Var(X) & = & E \left[(X-\mu)^2 \right] | ||
| + | & = & E(X^2) - \left[E(X) \right]^2 \\ | ||
| + | & = & \lambda^2 + \lambda - \lambda^2 \\ | ||
| + | & = & \lambda | ||
| + | \end{eqnarray*} | ||
mean_and_variance_of_poisson_distribution.1605892543.txt.gz · Last modified: 2020/11/21 02:15 by hkimscil