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. Thus X is a discrete random variable if

as u runs through the set of all possible values of the random variable X.

The Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution are among the most well-known discrete probability distributions.

If a random variable is discrete then the set of all possible values that it can assume is finite or countably infinite, because the sum of uncountably many positive real numbers (which is the smallest upper bound of the set of all finite partial sums) always diverges to infinity.