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Statistics · Statistical distributions

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Discrete random variables and probability distributions

What a discrete random variable is, and how a probability distribution lists each value it can take with the probability of that value — as a table or as a probability function. The key rule that the probabilities must sum to 1 (used to find an unknown), reading cumulative probabilities like P(X ≤ 2) off a distribution, and the special case of a discrete uniform distribution.

Statistics · Statistical distributions

Discrete random variables and probability distributions

What a discrete random variable is, and how a probability distribution lists each value it can take with the probability of that value — as a table or as a probability function. The key rule that the probabilities must sum to 1 (used to find an unknown), reading cumulative probabilities like P(X ≤ 2) off a distribution, and the special case of a discrete uniform distribution.

Why it works

A random variable is a quantity whose value is decided by chance — for example the score on a die, or the number of heads in three tosses. We write random variables with capital letters (XX, YY) and the particular values they take with small letters (xx). A discrete random variable can only take separate, listable values (usually whole-number counts).

A probability distribution is the complete list of the values XX can take together with the probability of each. It can be given as a table:

x1234P(X=x)0.10.30.40.2\begin{array}{c|cccc} x & 1 & 2 & 3 & 4 \\ \hline P(X = x) & 0.1 & 0.3 & 0.4 & 0.2 \end{array}

or as a probability function — a formula for P(X=x)P(X = x), e.g. P(X=x)=kxP(X = x) = kx for x=1,2,3,4x = 1, 2, 3, 4.

The one rule that governs everything. XX is certain to take some value, so the probabilities of all its values must add up to 11:

∑xP(X=x)=1.\sum_x P(X = x) = 1.

(Each probability also lies between 00 and 11.) This single fact lets you find a missing probability or an unknown constant kk: write down the sum, set it equal to 11, and solve.

Cumulative probabilities. From the distribution you can add the relevant rows to get things like P(X≤2)=P(X=1)+P(X=2)P(X \le 2) = P(X=1) + P(X=2), or P(X≥3)P(X \ge 3), or P(1<X≤3)P(1 < X \le 3). Match the inequality carefully — P(X<3)P(X < 3) does not include x=3x = 3, while P(X≤3)P(X \le 3) does.

The discrete uniform distribution. If every value is equally likely — like the score on a fair six-sided die — the distribution is discrete uniform, and each of the nn values has probability 1n\tfrac1n. For a fair die, P(X=x)=16P(X = x) = \tfrac16 for x=1,2,…,6x = 1, 2, \dots, 6.