# Two Different Formulas for Calculating the Expectation of a Discrete Random Variable and How to Read them

## A simple guide for keeping the symbology straight

There are two common formulas used to calculate the expectation (or expected value) of a random variable, depending on whether the random variable is discrete or continuous.

**Discrete Random Variable**

For a discrete random variable 𝑋 with a probability mass function (PMF) 𝑝(𝑥), the expected value 𝐸[𝑋] is given by the sum of the product of each possible value of the random variable and its probability:

where 𝑝(𝑥) represents the probability of *X* being equal to a specific value 𝑥.

## Alternative Formula with Explicit Random Variable Notation

In this formula, 𝑟*r* represents the possible values of 𝑋. The notation 𝑝(𝑋=𝑟)*p*(*X*=*r*) explicitly states the probability that *X* equals *r*. This form is more verbose and emphasizes that the probabilities are associated with the random variable *X* taking specific values *r*.

Both of these formulas compute the same quantity — the expected value of the random variable 𝑋 — using the sum of the products of values and their associated probabilities.