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We call the integral of a probability density function (PDF) a cumulative distribution function (CDF).

But what's the cumulative sum of a probability mass function (PMF) called? I've never heard the term "cumulative mass function" before, and the Wikipedia page for it redirects to the CDF page, so I'm confused what the proper terminology is.

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The proper terminology is Cumulative Distribution Function, (CDF). The CDF is defined as $$F_X(x) = \mathrm{P}\{X \leq x\}.$$ With this definition, the nature of the random variable $X$ is irrelevant: continuous, discrete, or hybrids all have the same definition. As you note, for a discrete random variable the CDF has a very different appearance than for a continuous random variable. In the first case, it is a step function; in the second it is a continuous function.

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