In statistics and probability, the **probability mass function**, abbreviated **pmf**, is a function that maps a discrete random variable to its probability. By convention, probability functions are denoted by lower case letters, often *f*. The pmf *f* for random variable *X *would be .

If *X* is a discrete random variable in a sample space *S*, then the probability mass function is defined as:

Like any probability function, the output of the pmf is between 0 and 1 inclusive, and the sum of the probability mass function’s image, ie., the probability of the sample space, is 1.

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