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Imagine rolling a die. The number that comes up is an example of a discrete random variable: you can list all the possible outcomes (1 through 6), and each outcome has a certain chance of occurring. In general, discrete random variables represent counts or measurements that can only take on specific values within a range.

Suppose we're talking about the chocolate chips in a cookie. If a cookie can have anywhere from 0 to 8 chocolate chips, each specific number of chips is a possible value of our discrete random variable. These potential values are not continuous; there can’t be 2.5 chips in a cookie—only whole numbers fit the frame of discrete data.

The expected value is a key term in probability, signifying the average outcome we expect after many trials of an experiment. For a discrete random variable, the expected value is calculated by multiplying each possible outcome by its probability and adding up all those products. This is like predicting the long-term average—how many chocolate chips you'd expect in a cookie if you had an unlimited number of cookies to sample. But remember, even if the expected value is a decimal, like 4.02 chips, a single cookie still won’t have a fractional chip; it's an average over many, many cookies.

The variance tells us how spread out the values of a random variable are. It is a measure of the variability or dispersion around the mean. To calculate the variance of a random variable, you need the expected value of the variable and the expected value of the square of the variable. The variance is then found by subtracting the square of the expected value from the expected value of the squared variable.

In the cookie example, a variance of 12.1 means that the number of chocolate chips is quite variable from one cookie to another. If the variance was closer to zero, we would expect nearly every cookie to have a chocolate chip count close to the mean.

The probability mass function (PMF) comes into play when we want to list the probabilities of each possible value of a discrete random variable. Like a recipe that tells you how much of each ingredient to add, a PMF tells you the likelihood of each possible number of chocolate chips in a cookie. In our case, the PMF is given by P(X=x), where X represents the variable (number of chips) and x is one of the possible values X can take. For example, P(X=3) is the probability that a randomly chosen cookie contains exactly three chocolate chips.

Understanding the PMF is crucial when we want to calculate metrics like mean and variance, as it clearly shows the probability of each event (or chocolate chip count) that can occur, and these form the basis for our statistical calculations.