A function that assigns a real number to each outcome in the sample space. 'Discrete' means it takes values from a countable set (like integers 0, 1, 2...).

A function that gives the probability that a discrete random variable is exactly equal to some value. Usually denoted as P(X=x) or p_X(x).

A function that returns the probability that the random variable will take a value less than or equal to x. Denoted as F_X(x) = P(X ≤ x). The function monotonically increases from 0 to 1.

The weighted average of all possible values of the random variable, where the weights are their probabilities. Represents the 'center' of the distribution. Denoted as E[X].

A measure of how spread out the values of the random variable are around its expectation. Calculated as the expected squared distance from the mean: V(X) = E[(X - E[X])²] or the working formula: E[X²] - (E[X])².

The square root of the variance. It returns the measure of dispersion to the same units as the original random variable. Denoted by the Greek letter sigma (σ).

A magical property of expectation! The expectation of a sum is equal to the sum of the expectations: E[aX + bY] = aE[X] + bE[Y]. This property is always true, even if the random variables are dependent.

A theoretical betting game (flipping a coin until getting 'tails' with a prize that doubles each time) where the expected payoff is infinite, but in reality, no reasonable person would pay a large amount to participate.