In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events.

### Cumulative distribution function

Because a probability distribution P on the real line is determined by the probability of a scalar random variable X being in a half-open interval (−∞, x], the probability distribution is completely characterized by its cumulative distribution function:

*F*(*x*) = *P*(*X ≤ x*)

for all *x *∈* R*

**Continuous distribution**

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

**Discrete distribution**

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

### Continuous distributions table

Distribution name | Distribution symbol | Probability density function (pdf) | Mean | Variance |

f_{x}(x) | μ = E(X) | σ^{2} = Var(X) | ||

Normal / gaussian | X ~ N(μ,σ^{2}) | μ | σ^{2} | |

Uniform | X ~ U(a,b) | (a+b)/2 | (b-a)^{2}/12 | |

Exponential | X ~ exp(λ) | 1/λ | 1/λ^{2} | |

Gamma | X ~ gamma(c, λ) | x>0, c>0, λ>0 | c/λ | c/λ^{2} |

Chi square | X ~ χ^{2}(k) | k | 2k | |

Wishart | ||||

F | X ~ F (k_{1}, k_{2}) | |||

Beta | ||||

Weibull | ||||

Log-normal | X ~ LN(μ,σ^{2}) | |||

Rayleigh | ||||

Cauchy | ||||

Dirichlet | ||||

Laplace | ||||

Levy | ||||

Rice | ||||

Student's t |

### Discrete distributions table

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) | np | np(1-p) | ||

Poisson | X ~ Poisson(λ) | λ ≥ 0 | λ | λ | |

Uniform | X ~ U(a,b) | (a+b)/2 | |||

Geometric | X ~ Geom(p) | p(1-p)^{k} | (1-p)/p | (1-p)/p^{2} | |

Hyper-geometric | X ~ HG(N,K,n) | N = 0,1,2,... K = 0,1,..,N n = 0,1,...,N | nK/N | ||

Bernoulli | X ~ Bern(p) | p | p(1-p) |