Probability Distribution

In probability and statistics distribution is a characteristic of a random variable, describes the probability of the random variable in each value.

Each distribution has a certain probability density function and probability distribution function.

Though there are indefinite number of probability distributions, there are several common distributions in use.

The probability distribution is described by the cumulative distribution function F(x),

which is the probability of random variable X to get value smaller than or equal to x:

F(x) = P(X ≤ x)

The cumulative distribution function F(x) is calculated by integration of the probability density function f(u) of continuous random variable X.

The cumulative distribution function F(x) is calculated by summation of the probability mass function P(u) of discrete random variable X.

Continuous distribution is the distribution of a continuous random variable.

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Distribution name	Distribution symbol	Probability density function (pdf)	Mean	Variance
		f_X(x)	μ = E(X)	σ²= Var(X)
Normal / gaussian	X ~ N(μ,σ²)	$\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$	μ	σ²
Uniform	X ~ U(a,b)	$\begin{Bmatrix}\frac{1}{b-a} & ,a\leq x\leq b\\ & \\0 & ,otherwise\end{matrix}$		$\frac{(b-a)^2}{12}$
Exponential	X ~ exp(λ)	$\begin{Bmatrix}\lambda e^{-\lambda x} & x\geq 0\\ 0 & x<0\end{matrix}$	$\frac{1}{\lambda}$	$\frac{1}{\lambda^2}$
Gamma	X ~ gamma(c, λ)	$\frac{\lambda ^c x^{c-1}e^{-\lambda x}}{\Gamma (c)}$ x > 0, c > 0, λ > 0	$\frac{c}{\lambda }$	$\frac{c}{\lambda ^2}$
Chi square	X ~ χ²(k)	$\frac{x^{k/2-1}e^{-x/2}}{2^{k/2}\Gamma (k/2)}$	k	2k
Wishart
F	X ~ F (k₁, k₂)
Beta
Weibull
Log-normal	X ~ LN(μ,σ²)
Rayleigh
Cauchy
Dirichlet
Laplace
Levy
Rice
Student's t

Discrete distribution is the distribution of a discrete random variable.

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Distribution name	Distribution symbol	Probability mass function (pmf)		Mean	Variance
		f_x(k) = P(X=k) k = 0,1,2,...		E(x)	Var(x)
Binomial	X ~ Bin(n,p)	$\binom{n}{k}p^{k}(1-p)^{n-k}$		np	np(1-p)
Poisson	X ~ Poisson(λ)		λ ≥ 0	λ	λ
Uniform	X ~ U(a,b)	$\begin{Bmatrix}\frac{1}{b-a+1} & ,a\leq k\leq b\\ & \\0 & ,otherwise\end{matrix}$		$\frac{a+b}{2}$	$\frac{(b-a+1)^{2}-1}{12}$
Geometric	X ~ Geom(p)	$p(1-p)^{k}$		$\frac{1-p}{p}$	$\frac{1-p}{p^2}$
Hyper-geometric	X ~ HG(N,K,n)		N = 0,1,2,... K = 0,1,..,N n = 0,1,...,N	$\frac{nK}{N}$	$\frac{nK(N-K)(N-n)}{N^2(N-1)}$
Bernoulli	X ~ Bern(p)	$\begin{Bmatrix}(1-p) & ,k=0\\ p & ,k=1\\ 0 & ,otherwise\end{matrix}$		p	p(1-p)