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.
Introduction A probability distribution is a mathematical description of the probabilities of events, i.e. subsets of the sample space. The sample space, often represented in notation by Ω , {\displaystyle \ \Omega \ ,} is the set of all possible outcomes of a random phenomenon being observed. The sample space may be any set of numbers, vectors, labels, or whatever else. For example, the sample space of a coin flip could be Ω = {"heads", "tails"} , whereas for a die roll, it could be Ω = {1, 2, 3, 4, 5, 6} . To define probability distributions for the specific case of random variables (so that the sample space can mapped to a measurable space, for example the real numbers), it is common to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function p {\displaystyle p} assigning a probability to each possible outcome (e.g. when throwing a fair die, each of the six digits "1" to "6", corresponding to the number of dots on the die, has probability 1 6 {\displaystyle {\tfrac {1}{6}}} of being on top when it lands). The probability of an event is then defined to be the sum of the probabilities of all outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is
P ( 2 ) + P ( 4 ) + P ( 6 ) = 1 6 + 1 6 + 1 6 = 1 2 {\displaystyle P(2)+P(4)+P(6)={\frac {1}{6}}+{\frac {1}{6}}+{\frac {1}{6}}={\frac {1}{2}}}
In contrast, when a random variable takes values from a continuum, then unless the probability density function has any infinitely-dense peaks, any individual outcome has probability zero. For such continuous random variables, only events that include infinitely many outcomes, such as intervals, have probability greater than 0. For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale can provide arbitrarily many digits of precision. Then, the probability that it weighs exactly 500 g must be zero because no matter how high the level of precision chosen, it cannot be assumed that there are no non-zero digits after those output by the scale. However, for the same use case, it is possible to meet quality control requirements such as that a package of "500 g" of ham must weigh between 490 g and 510 g. This is possible because this measurement does not require infinite precision from the underlying equipment, and it provides some tolerance for variability in physical objects and processes.
Continuous probability distributions can be described by means of the cumulative distribution function, which describes the probability that the random variable is no larger than a given value (i.e., P(X ≤ x) for some x. The cumulative distribution function is the area under the probability density function from -∞ to x, as shown in figure 1. Most continuous probability distributions encountered in practice are not only continuous but also absolutely continuous. Such distributions can be described by their probability density function. Informally, the probability density f {\displaystyle f} of a random variable X {\displaystyle X} describes the relative value of the infinitesimal probability that X {\displaystyle X} takes any value x {\displaystyle x} — that is P ( x ≤ X < x + Δ x ) ≈ f ( x ) Δ x {\displaystyle P(x\leq X<x+\Delta x)\approx f(x)\,\Delta x} as Δ x > 0 {\displaystyle \Delta x>0} becomes arbitrarily small. The probability that X {\displaystyle X} lies in a given interval can be computed rigorously by integrating the probability density function over that interval.
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