Probability distribution

In probability theory and statistics, a probability distribution describes how probabilities are assigned to the possible results of a random phenomenon—more precisely, to events, which are sets of possible outcomes of a probabilistic experiment. Informally, a probability distribution tells us how likely different results are. Formally, it is a probability measure: a function that assigns probabilities to events in a way that satisfies the axioms of probability.

Probability distributions are closely linked to random variables. A random variable is a function that assigns a value to each outcome of a probabilistic experiment; it induces a probability distribution on the set of values it can take. For example, the result of a coin toss can be represented by a random variable X that equals 1 for heads and 0 for tails. If the coin is fair, this distribution assigns probability 1/2 to X = 1 and probability 1/2 to X = 0. Viewed as a probability measure, the distribution of X assigns ℙ(X ∈ A) to each set A ⊆ {0,1}; for a fair coin, ℙ(X ∈ {1}) = ℙ(X ∈ {0}) = 1/2, ℙ(X ∈ {0,1}) = 1, and ℙ(X ∈ ∅) = 0.

In practice, probability distributions are often described by functions such as cumulative distribution functions, probability mass functions, or probability density functions. Which description is used depends on the nature of the distribution: probability mass functions are used for discrete distributions, while probability density functions are used for many continuous distributions.

Probability distributions that occur frequently or have special theoretical importance are often given specific names; examples are collected in the list of probability distributions.