Probability distributions

Main distribution examples:

$\begin{array}{c|c|c|clcr}\text{Distribution} & \text{PDF or PMF} & \text{Mean} & \text{Variance}\\ \hline Bernoulli(p)& \begin{cases} p, & \text{if $x=1$} \\ 1-p & \text{if $x=0$} \end{cases} & p & p(1-p)\\ \hline Binomial(n,p) & \binom{n}{k}p^k(1-p)^{n-k} \text{ for } k = 0,1,....n& np & np(1-p)\\ \hline Geometric(p) & p(1-p)^{k-1} \text{ for } k = 0,1, 2...& \dfrac{1}{p} & \dfrac{1-p}{p^2}\\ \hline Poisson(\lambda) & \dfrac{e^{- \lambda}\lambda^k}{k!} \text{ for } k = 0,1,2...& \lambda & \lambda\\ \hline Uniform(a,b) & \dfrac{1}{b-a} \text{ for all } x \in (a,b)& \dfrac{a+b}{2} & \dfrac{(b-a)^2}{12}\\ \hline Gaussian(\mu,\sigma) & \dfrac{1}{\sigma \sqrt{2\pi}}e^{-\dfrac{(x- \mu)^2}{2\sigma^2}} \text{ for all } x \in (-\infty,\infty)& \mu & \sigma^2\\ \hline Exponential(\lambda) & \lambda e^{-\lambda x} \text{ for all } x \ge 0, \lambda \ge 0& \dfrac{1}{\mu} & \dfrac{1}{\mu^2}\\ \end{array}$

PDF: Probability Desity Funcion
PMF: Probability Mass Function

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