Discrete random variable

A discrete random variable takes values you could write down in a list, with a gap between each possible value and the next. Every value carries its own probability $P(X = x)$, which is a genuine positive number rather than zero, and those probabilities total 1 across the list. Counting is the usual source: how many, how often, how many out of $n$.

Let $X$ be the number of heads in 3 flips of a fair coin. There are 8 equally likely sequences. TTT gives $X = 0$; HTT, THT and TTH give $X = 1$; HHT, HTH and THH give $X = 2$; and HHH gives $X = 3$. So the probabilities are $\frac{1}{8}$, $\frac{3}{8}$, $\frac{3}{8}$ and $\frac{1}{8}$, which add to $\frac{8}{8} = 1$, and $P(X = 2) = 0.375$.

"Discrete means whole numbers" is the misreading, and it is wrong. A scratch card paying 0, 2.50 or 10 dollars is discrete. So is a shoe size that steps 7, 7.5, 8. What makes a variable discrete is that its values sit apart from each other with nothing in between, not that they happen to be integers.

Discrete does not mean finite either. Count how many cars pass before the first red one and the possible values run 0, 1, 2 and onward without end, yet they are still listable and their probabilities still total 1. The real dividing line is against continuous variables. For those, $P(X = x) = 0$ at every single value and probability exists only over intervals, which is why a weight rounded to the nearest tenth of a kilogram is a discrete record of a quantity that is not.

That difference decides the machinery you use. A discrete variable gets a table of probabilities, a probability histogram, and sums; a continuous one gets a density curve and areas. Discrete random variables enter at topic 2.8 of Unit 2, and their mean and standard deviation at topic 2.9, Parameters of Random Variables.