Random Variables, PMF & CDF

Toss three coins and count how many come up heads. The answer is a number — 0, 1, 2, or 3 — and that small act, of turning a messy outcome like heads-tails-heads into a single tidy number, is something you do without thinking. Give the act a name and it becomes the workhorse of everything that follows in probability.

Naming the act

A rule that turns each outcome of an experiment into a number is a random variable. Counting heads is one; so is the score on a die, or the time you wait for a bus. When the numbers sit on a separated, countable list — 0, 1, 2, 3 heads — the variable is discrete, and that is this lesson. When they can land anywhere in a stretch — any waiting time at all — it is continuous, which waits for the next lesson.

Two companions describe a discrete random variable completely: a list of its chances, and a running total of them.

The list and the running total

The list is the probability mass function, p(x) = P(X = x) — the chance of each individual value. For it to make sense, two things must hold, and both are just the axioms from before: no chance is negative, and the chances over all the values add up to one whole.

p(x) ≥ 0   for every x        and        Σ p(x) = 1

The running total is the cumulative distribution function, F(x) = P(X ≤ x) — the chance of landing at x or anywhere below it. Because it only ever adds non-negative masses as x moves right, the running total can climb or stay level but never fall. And for a discrete variable it climbs in jumps: it leaps upward by p(x) exactly at each value, then runs flat until the next one. Drawn out, it is a staircase.

That staircase shape carries the properties GATE tests, and a 2025 MSQ asked exactly which of them hold for every CDF. They follow from “running total of non-negative masses”: F is non-decreasing, it is right-continuous (at a jump it already sits at the higher value), and it runs from F(−∞) = 0 up to F(+∞) = 1.

Reading chances off the staircase

Once you have the staircase, one identity does most of the work. The chance of landing in an interval is the height climbed across it — the total at the top minus the total already counted at the bottom:

P(a < X ≤ b) = F(b) − F(a)

And a single jump is a single mass: P(X = x) = F(x) − F(x⁻), the height of the step at x.

Watching it built, value by value

A discrete random variable X takes the values 0, 1, 2, 3 with masses p = 0.1, 0.3, 0.4, 0.2. Find the CDF, then P(X ≤ 2) and P(1 < X ≤ 3).

First check it is a real PMF: every mass is non-negative, and 0.1 + 0.3 + 0.4 + 0.2 = 1. Good. Now accumulate left to right, each entry the running total so far:

x         0      1      2      3
p(x)     0.1    0.3    0.4    0.2
F(x)     0.1    0.4    0.8    1.0      ← running total

P(X ≤ 2)     = F(2)              = 0.8
P(1 < X ≤ 3) = F(3) − F(1)       = 1.0 − 0.4 = 0.6

P(X ≤ 2) is simply the staircase’s height at 2, the running total 0.1 + 0.3 + 0.4. The interval P(1 < X ≤ 3) leaves out X = 1 but keeps X = 3, so it is the masses at 2 and 3, 0.4 + 0.2 = 0.6 — which is exactly F(3) − F(1).

A question to carry forward

A full PMF table — every value and its mass — is more than you usually want to carry around. Here is the thread for the next lesson: most of the time two numbers would do, one saying roughly where the variable sits and one saying roughly how much it swings. How do we boil a whole table down to those two summary numbers?

In one breath

• A random variable turns each outcome into a number; discrete = a countable list of values (this lesson), continuous = a range (next).
• PMF p(x) = P(X=x): valid iff p(x) ≥ 0 and Σ p(x) = 1.
• CDF F(x) = P(X ≤ x) = running total: non-decreasing, right-continuous, F(−∞)=0, F(+∞)=1; a staircase for a discrete RV, jumping by p(x) at each value.
• Interval chance: P(a < X ≤ b) = F(b) − F(a); a single jump is the mass P(X=x) = F(x) − F(x⁻).
• The 2025 MSQ trap: a CDF is right-continuous and non-decreasing — never left-continuous, never decreasing.

Practice