Probability Distribution

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 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.

F(x)=P(X\leq x)=\int_{-\infty }^{x}f(u)du

Discrete distribution


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

F(x)=P(X\leq x)=\sum_{-\infty }^{x}P(u)du

Continuous distributions table

Discrete distributions table