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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.

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(Xx)

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 nameDistribution symbolProbability density function (pdf)MeanVariance
  

fX(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}}μσ2
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}
ExponentialX ~ exp(λ)\begin{Bmatrix}\lambda e^{-\lambda x} & x\geq 0\\ 0 & x<0\end{matrix}\frac{1}{\lambda}\frac{1}{\lambda^2}
GammaX ~ 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 distributions table

Discrete distribution is the distribution of a discrete random variable.

Discrete distribution example

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Discrete distributions table

Distribution nameDistribution symbolProbability mass function (pmf)MeanVariance
  fx(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)