In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. A probability mass function differs from a probability density function in that the values of the latter, defined only for continuous random variables, are not probabilities; rather, its integral over a set of possible values of the random variable is a probability. Probability theory is the mathematical study of probability. ... In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... By one convention, a random variable X is called continuous if its cumulative distribution function is continuous. ...

Mathematical description

Suppose that X is a discrete random variable, taking values on some countable sample space  S ⊆ R. Then the probability mass function  f_X(x)  for X is given by In mathematics the term countable set is used to describe the size of a set, e. ... In probability theory, the sample space, often denoted S, Ω or U (for universe), of an experiment or random trial is the set of all possible outcomes. ...

Note that this explicitly defines  f_X(x)  for all real numbers, including all values in R that X could never take; indeed, it assigns such values a probability of zero. (Alternatively, think of  Pr(X = x)  as 0 when  x ∈ RS.) In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...

The discontinuity of probability mass functions reflects the fact that the cumulative distribution function of a discrete random variable is also discontinuous. Where it is differentiable (i.e. where x ∈ RS) the derivative is zero, just as the probability mass function is zero at all such points.

Examples

A simple example of a probability mass function is the following. Suppose that X is the outcome of a single coin toss, assigning 0 to tails and 1 to heads. The probability that X = x is just 0.5 on the state space {0, 1} (this is a Bernoulli random variable), and hence the probability mass function is In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist James Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ...

Probability mass functions may also be defined for any discrete random variable, including constant, binomial (including Bernoulli), negative binomial, Poisson, geometric and hypergeometric random variables. In probability theory, a constant random variable is a discrete random variable that takes a constant value, regardless of any event that occurs. ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist James Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ... In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before... In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. ...