The cumulative distribution function

In the last lesson, we learned that the PMF gives $P(X = k)$ , the probability of exactly $k$ . But sometimes we want to know: "What's the probability of getting at most $k$ ?"

Defining the CDF

Cumulative Distribution Function

The CDF of a random variable $X$ is: $F_X(x) = P(X \leq x)$

It accumulates probability from left to right.

Building a CDF Step by Step

Step through each value to see how the CDF accumulates probability from left to right.

Distribution

PMF — P(X = k)

0.125

0.375

0.125

CDF — F(x) = P(X ≤ x)

Press Next to add the first probability bar.

0 / 4

The CDF tells you: "What fraction of outcomes are at or below this value?"

If F(3) = 0.7, what's P(X ≤ 3)? (decimal, e.g. 0.42)

If F(3) = 0.7, what's P(X > 3)? (decimal, e.g. 0.42)

Properties of the CDF

The CDF always:

Starts at 0: No probability to the left of the smallest value
Ends at 1: All probability is accounted for as $x \to \infty$
Never decreases: Probability only accumulates

For discrete variables, the CDF looks like a staircase, jumping up at each possible value.

The skyscraper plot

The PMF bars stack up to form the CDF staircase.

The Skyscraper Plot

PMF

0.10

0.15

0.25

0.15

0.07

0.03

CDF

0123456

Mean

2.53

Median

Mode

Var

2.23

Each step in the CDF equals the height of the corresponding PMF bar.

Build a PMF

PMF

CDF

Sum: 1.00Each step = bar height

Draw any PMF (the bars), and the CDF (the staircase) automatically follows. The height of each step equals the probability mass at that point.

Converting between PMF and CDF

You can go either direction:

From PMF to CDF: Sum up the probabilities $F_X(k) = \Large\sum_{j \leq k} p_X(j)$

From CDF to PMF: Take the difference $p_X(k) = F_X(k) - F_X(k-1)$

If p(1)=0.2, p(2)=0.3, p(3)=0.5, what is F(2)? (decimal, e.g. 0.42)

If F(4)=0.8 and F(5)=0.95, what is p(5)? (decimal, e.g. 0.42)

Probability of an interval

The CDF makes interval calculations easy:

Probability of an Interval

$P(a < X \leq b) = F_X(b) - F_X(a)$

If F(5)=0.8 and F(2)=0.3, what is P(2 < X ≤ 5)? (decimal, e.g. 0.42)

The median and quantiles

The CDF helps us find important summary statistics.

Median

The median is the smallest value $m$ such that $F_X(m) \geq 0.5$ . Half the probability is at or below the median.

More generally, the $p$ -th quantile (or percentile) is the smallest $x$ where $F_X(x) \geq p$ .

Quantile Finder

Probability: 50%

10%99%

Quantile at p = 0.50

Smallest x where F(x) ≥ p

What percentile is the median? (whole number)

If F(5)=0.45 and F(6)=0.52, what is the median? (whole number)

Summary

Concept	Symbol	What it tells you
PMF	$p_X(k)$	Probability of exactly $k$
CDF	$F_X(x)$	Probability of at most $x$
Median	$m$	50th percentile

When you see a probability question, ask: "Do I need an exact value (use PMF) or a cumulative probability (use CDF)?" This guides your approach.

What's next

Now that we can describe random variables with PMFs and CDFs, let's meet some famous distributions. Next up: the Bernoulli and Binomial distributions.