| Discrete |
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| Poisson(\(\mu\)) |
\(x = 0, 1, 2, \ldots\) |
\(e^{-\mu}\frac{\mu^x}{x!}\) |
No closed form |
\(\mu\) |
\(\sqrt{\mu}\) |
| Binomial(\(n\), \(p\)) |
\(x = 0, 1, 2, \ldots, n\) |
\(\binom{n}{x}p^x(1-p)^{n-x}\) |
No closed form |
\(np\) |
\(\sqrt{np(1-p)}\) |
| Geometric(\(p\)) |
\(x = 1, 2, 3, \ldots\) |
\(p(1-p)^{x-1}\) |
\(1-(1-p)^x\) |
\(\frac{1}{p}\) |
\(\sqrt{\frac{1-p}{p^2}}\) |
| Negative Binomial(\(r\), \(p\)) |
\(x = r, r+1, r+2, \ldots\) |
\(\binom{x-1}{x-r}p^r(1-p)^{x-r}\) |
No closed form |
\(\frac{r}{p}\) |
\(\sqrt{\frac{r(1-p)}{p^2}}\) |
| Hypergeometric(\(n\), \(N_1\), \(N_0\)); \(N = N_1+N_0, p=\frac{N_1}{N}\) |
\(x = 0, 1, \ldots, n; x\le N_1; n-x\le N_0\) |
\(\frac{\binom{N_1}{x}\binom{N_0}{n-x}}{\binom{N_1 + N_0}{n}}\) |
No closed form |
\(np\) |
\(\sqrt{np(1-p)\left(\frac{N-n}{N-1}\right)}\) |
| Continuous |
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| Uniform(\(a\), \(b\)) |
\(a<x<b\) |
\(\frac{1}{b-a}\) |
\(\frac{x-a}{b-a}\) |
\(\frac{a+b}{2}\) |
\(\frac{b-a}{\sqrt{12}}\) |
| Exponential(\(\lambda\)) |
\(x>0\) |
\(\lambda e^{-\lambda x}\) |
\(1-e^{-\lambda x}\) |
\(\frac{1}{\lambda}\) |
\(\frac{1}{\lambda}\) |
| Normal(\(\mu\), \(\sigma\)) |
\(-\infty < x< \infty\) |
\(\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)\) |
No closed form |
\(\mu\) |
\(\sigma\) |
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