Random variables and their distributions

A number attached to an outcome

A random variable is a numerical outcome of a random experiment — a rule that assigns a number to each outcome in the sample space. Capital $X$ names the variable ("the number of late invoices in a batch"); lower-case $x$ names one value it might take ($x=2$). $P(X=2)$ reads "the probability that the number of late invoices is 2".

A discrete random variable takes separated values you can list — counts, almost always: $0,1,2,\dots$. A continuous one takes any value in an interval — a time, a weight, a dollar amount — and is Module 7's subject.

The probability distribution

The probability mass function (pmf) lists each value with its probability. Two conditions make a list a valid pmf:

1. every probability lies between 0 and 1: $0\le p(x)\le 1$;
2. the probabilities sum to one: $\sum p(x)=1$.

Worked example — Maple Ledger's batches of three

Maple Ledger processes invoices in batches of three. Let $X$ = the number of invoices in a batch that are paid late. From a year of records:

The second condition fills the gap: $p(3)=1-(0.40+0.35+0.15)=0.10$. Every entry is between 0 and 1, so the pmf is valid. A list such as 0.4, 0.4, 0.4 fails condition 2 (sum 1.2); a list containing −0.1 fails condition 1 however its total behaves.

Translating the phrase

The paper asks in words; the pmf answers in inequalities. With the table above:

The boundary is the whole difficulty. At least and at most include the number; more than and fewer than exclude it. "No more than 1" is "at most 1"; "at least 1" is $1-p(0)=0.60$ — Module 5's complement again. Write the inequality before touching the table, every time; on a binomial question with twenty terms that habit is the difference between one subtraction and nineteen additions.

Cumulative probability

$P(X\le k)$ — the running total of the pmf up to $k$ — is the cumulative probability. For the batch: $P(X\le 0)=0.40$, $P(X\le 1)=0.75$, $P(X\le 2)=0.90$, $P(X\le 3)=1$. Every phrase in the table is a cumulative value, or one minus a cumulative value, or a difference of two. R's pbinom and ppois (L6.5, L6.6) return exactly this running total, which is why the phrase translation matters twice.