# 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 distributions table

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

fX(x)

μ = E(X)

σ2 = Var(X)

X ~ N(μ,σ2)

μ σ 2
Uniform

X ~ U(a,b)

Exponential X ~ exp(λ)
Gamma X ~ gamma(c, λ)

x > 0, c > 0, λ > 0

Chi square

X ~ χ 2(k)

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 distributions table

Distribution name Distribution symbol Probability mass function (pmf) Mean Variance
fx(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,...,N

Bernoulli

X ~ Bern(p)

p

p(1-p)

Currently, we have around 5612 calculators, conversion tables and usefull online tools and software features for students, teaching and teachers, designers and simply for everyone.

There are many very usefull online free tools and we would be happy if you share our page to others or send us any suggestions for other tools which will come on your mind. Also in case you find any of our tools that it does not work properly or need a better translation - please let us know. Our tools will make your life easier or simply help you to do your work or duties faster and in more effective way.

These below are the most commonly used by many users all over the world.

And we are still developing more. Our goal is to become the one-stop, go-to site for people who need to make quick calculations or who need to find quick answer for basic conversions.

Additionally, we believe the internet should be a source of free information. Therefore, all of our tools and services are completely free, with no registration required. We coded and developed each calculator individually and put each one through strict, comprehensive testing. However, please inform us if you notice even the slightest error – your input is extremely valuable to us. While most calculators on Justfreetools.com are designed to be universally applicable for worldwide usage, some are for specific countries only.

Page Id: 1610