A discrete random variable is any random variable that can only take on integer values, for example the number of coin tosses before getting a head. ## Expectation The expectation of the discrete random variable is the probability-weighted sum of its values. The expectation is the same as the average value if all outcomes are equally likely. Note that the expectation may not be a valid value from the random variable. For example a binomial random variable will spit out either 1 or 0, but its expectation is 0.5. $E(X) = \sum_{k=1}^\infty kP(X=k)$ ## Variance The variance of a discrete random variable is the sum of squared deviations from the mean. Variance increases as sample size increases simply because there are more deviations to sum. You would expect more variance when flipping a coin 10 times than 5 times. $V(X) = E(X^2) - (E(X))^2$ The first term $E((X^2)$ is called the "second moment". The calculation of the second moment is $E(X^2) = \sum_{k=1}^\infty k^2 P(X=k)$ Related: the [[standard deviation]] is the square root of the variance and the [[standard error of the mean]] is the standard deviation divided by the number of samples taken.