Summary of Some Common Distributions

Name and parameters Possible values pmf/pdf cdf Mean SD
Discrete
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
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 \(\mu\) \(\sigma\)