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.