Probability distribution

A probability distribution is a function that gives the Probability that a Random variable will take on a specific value or set of values on measurement. If a probability distribution accurately describes the statistical behavior of a random variable, it is said that the variable follows that distribution.

The term "probability distribution" is sometimes used to refer to the image of the distribution function. For example, say that a loaded coin has a 40% chance to land as heads and 60% as tails, then "probability distribution" can refer to:

• The function $p:\{ \text{Heads},\text{Tails} \}\to[0,1]$ that maps $p(\text{Heads})=0.4$ and $p(\text{Tails})=0.6$. This is the formally correct definition.
• The image of $p$, $\{ 0.4, 0.6 \}$, which contains the two probabilities, leaving the mapping implied. This isn't correct, but it is sometimes used.

Probability distributions may be discrete or continuous.

• Discrete probability distributions are known as probability mass functions.
• Continuous probability distributions are known as probability density functions. However, due to complexities arising from the infinite outcomes over a continuous sample space, the distribution is only fully defined when the cumulative distribution function is also given.

Regardless of the type, most distribution functions have parameters. These are numerical values that modify the shape and behavior of the distribution and are chosen separately from the usual arguments. For example, the Gaussian distribution takes one real number $x$ as a regular argument and has two parameters: the mean $\mu$ and the variance $\sigma ^{2}$. To distinguish the two, it is typical to separate regular arguments from parameters with a semicolon. The Gaussian probability density function is commonly denoted $f(x;\mu,\sigma ^{2})$.