## Probability Distribution

In probability and statistics

distributionis 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.

## Cumulative distribution function

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)## 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

Continuous distribution is the distribution of a continuous random variable.

## Continuous distribution example

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## 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})μσ^{ 2}Uniform

X~U(a,b)Exponential X~exp(λ)Gamma X~gamma(c, λ)

x> 0,c> 0, λ > 0Chi square

X~χ^{ 2}(k)

k2

kWishart 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

Discrete distribution is the distribution of a discrete random variable.

## Discrete distribution example

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## 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)Geometric

X~Geom(p)

Hyper-geometric

X~HG(N,K,n)

N= 0,1,2,...

K= 0,1,..,N

n= 0,1,...,NBernoulli

X~Bern(p)

p

p(1-p)

## See also