mean_and_variance_of_geometric_distribution
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| mean_and_variance_of_geometric_distribution [2023/10/17 20:32] – hkimscil | mean_and_variance_of_geometric_distribution [2023/10/18 17:33] (current) – hkimscil | ||
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| (1-(1-p))E(X) & = \underline{p} \left[1 + \underline{(1-p)} + \underline{(1-p)^2} + \underline{(1-p)^3} + \underline{(1-p)^4} + . . . \right] \\ | (1-(1-p))E(X) & = \underline{p} \left[1 + \underline{(1-p)} + \underline{(1-p)^2} + \underline{(1-p)^3} + \underline{(1-p)^4} + . . . \right] \\ | ||
| & = \sum_{k=0}^{\infty} \left(p \cdot (1-p)^{k} \right) \\ | & = \sum_{k=0}^{\infty} \left(p \cdot (1-p)^{k} \right) \\ | ||
| - | (1-(1-p))E(X) & = p \frac {1 - (1-p)^k} {1-(1-p)}, \;\;\; \text{because | + | (1-(1-p))E(X) & = p \frac {1 - (1-p)^k} {1-(1-p)}, \;\;\; \text{because |
| & = p \frac {1}{1-(1-p)} \\ | & = p \frac {1}{1-(1-p)} \\ | ||
| p \cdot E(X) & = 1 \\ | p \cdot E(X) & = 1 \\ | ||
| Line 41: | Line 41: | ||
| ====== Variance ====== | ====== Variance ====== | ||
| + | For (1), see | ||
| + | [[Expected value and variance properties# | ||
| \begin{align} | \begin{align} | ||
| Var(X) & = E((X-E(X))^{2}) \nonumber \\ | Var(X) & = E((X-E(X))^{2}) \nonumber \\ | ||
| Line 65: | Line 67: | ||
| \end{align} | \end{align} | ||
| - | 위에서 $ (6) - (7) = $ 은 | + | 위에서 $ (7) - (8) = $ 은 |
| \begin{align} | \begin{align} | ||
| Line 92: | Line 94: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | 따라서 $(5)$ 는 | + | 따라서 $(9)$ 는 |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| E(X^{2}) | E(X^{2}) | ||
| Line 108: | Line 110: | ||
| 이는 기하분포의 평균에서 p가 생략된 것과 같다. | 이는 기하분포의 평균에서 p가 생략된 것과 같다. | ||
| \begin{align*} | \begin{align*} | ||
| - | \left(E(X) = \right) | + | E(X) = \sum_{k=1}^{n} k (1-p)^{k-1} \cdot p & = \frac{1}{p} \\ |
| \sum_{k=1}^{n} k (1-p)^{k-1} & = \frac{1}{p^2} \\ | \sum_{k=1}^{n} k (1-p)^{k-1} & = \frac{1}{p^2} \\ | ||
| \\ | \\ | ||
| Line 118: | Line 120: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | n이 무한대로 간다고 생각하면 (, [[: | + | n이 무한대로 간다고 생각하면 ([[: |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| \sum_{k=1}^{\infty} ar^{k-1} & = & a \frac{1-r^n}{1-r} \\ | \sum_{k=1}^{\infty} ar^{k-1} & = & a \frac{1-r^n}{1-r} \\ | ||
| a=1; r = (1-p); \\ | a=1; r = (1-p); \\ | ||
| \sum_{k=1}^{\infty} (1-p)^{k-1} & = & \frac{1-(1-p)^n}{1-(1-p)} \\ | \sum_{k=1}^{\infty} (1-p)^{k-1} & = & \frac{1-(1-p)^n}{1-(1-p)} \\ | ||
| - | (1-p)^n | + | (1-p)^n |
| & = & \frac{1}{p} | & = & \frac{1}{p} | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| Line 132: | Line 134: | ||
| \\ | \\ | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | |||
| + | 따라서 | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
mean_and_variance_of_geometric_distribution.1697542337.txt.gz · Last modified: 2023/10/17 20:32 by hkimscil