Cumulative Distribution Function

The cumulative distribution function, denoted $F(x)$, gives the probability that a random variable takes a value less than or equal to a given number $x$:

The c.d.f. is defined identically in spirit for both discrete and continuous random variables, though it looks different in each case.

c.d.f. of a discrete random variable

For a discrete random variable, $F(x)$ is obtained by adding up the probabilities of every value up to and including $x$:

The result is a step function — it jumps up by $p_i$ at each possible value $x_i$ and stays flat in between.

Illustration. Take the distribution:

The c.d.f. is built by accumulating probabilities from the left:

Notice $F(4) = 1$, since $X$ is certain to be at most its largest value.

c.d.f. of a continuous random variable

For a continuous random variable, the sum is replaced by an integral of the p.d.f.:

Here $F(x)$ is a smooth (not stepped) function, and the p.d.f. can be recovered back from it by differentiation: $f(x) = F'(x)$.

Key properties of any c.d.f.

• $0 \leq F(x) \leq 1$ for every $x$ — it is a probability, so it can never go below $0$ or above $1$.
• $F(x)$ is non-decreasing — as $x$ increases, $F(x)$ never falls; it only stays the same or rises.
• $F(x) \to 0$ as $x \to -\infty$, and $F(x) \to 1$ as $x \to \infty$. …