what is pmf in statistics

In statistics, a probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. It characterizes the distribution of a discrete random variable and provides the possible values and their associated probabilities. The PMF is defined as a function over the sample space of a discrete random variable X, which gives the probability that X is equal to a certain value. The PMF satisfies two conditions: (i) for all x in the sample space, f(x) is never negative, and (ii) the sum of f(x) over the entire sample space will always be 1. The PMF is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete. The PMF differs from a probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables. The PMF is used in binomial and Poisson distribution to find the probability value where it uses discrete values.