Foundations — Cumulative distribution functions

Before you can trust that one sentence, you need the vocabulary hidden inside it. This page builds every symbol the parent CDF note uses, from absolute zero, in the order that each one depends on the last. Nothing is used before it is drawn.

1. Probability $P(\text{something})$ — a number between 0 and 1

The picture. Imagine a bar completely filled with paint. The paint is all the certainty in the world — it sums to $1$. Any event grabs a slice of that bar. A big slice = likely; a thin sliver = unlikely.

Why the topic needs it. Everything on this page is ultimately a number produced by $P(\dots )$. The running total we are chasing will just be a particular probability, so we must first be comfortable that every $P$ lives on a $[0,1]$ ruler. We cannot yet write the CDF's defining formula — it needs the symbols built in sections 2 and 3 first.

2. Random variable $X$ — a number decided by chance

The picture. Think of $X$ as a labelled dial that a random spin will land somewhere on. Rolling a die: $X$ is the face that shows up. Measuring where a bullet lands: $X$ is the distance in metres.

See Random Variables for the full treatment.

3. The symbol $\le$ — "at or below"

Now — with $P$, $X$, $x$ and $\le$ all defined — we can finally read the event $X\le x$: the random outcome $X$ turned out to be less than or equal to the fixed number $x$.

The picture. On a number line, $X\le x$ means $X$ landed anywhere in the shaded region to the left of (and including) the marker $x$.

4. The CDF $F_X(x)$ itself — assembling the pieces

Every ingredient is now on the table: a probability $P$, a random machine $X$, a fixed plug-in value $x$, and the event $X\le x$. Snap them together.

Why $P(X\le x)$ and not $P(X=x)$ or $P(X<x)$? We want a running total, not a single spot. Using $\le$ (the "or equal" bar) means "everything up to and including $x$." That choice is exactly what makes the CDF right-continuous (section 9) and what makes a die's CDF jump at each face value. Every subtlety in the parent note traces back to this one bar.

5. Two flavours of randomness: discrete vs continuous

The picture. Discrete = a row of separate poles you can stand on. Continuous = a smooth ramp you can stand anywhere on.

6. PMF $p_X(x)$ — probability in lumps (discrete)

The picture. A row of vertical sticks, one per allowed value; the height of each stick is its probability. For a fair die every stick has height $\frac{1}{6}$.

Why the topic needs it. The discrete CDF is literally the running sum of these lumps: $F_X(x)={\sum }_{x_i\le x}{p}_X(x_i)$. Section 7 explains that $\sum$ symbol.

7. The sum symbol $\sum$ — "add up all of these"

Worked read-aloud (fair die, $F_X(3.7)$). The values $\le 3.7$ are $1,2,3$. So the sum is

That is exactly the parent note's staircase example — now every symbol in it is earned.

8. PDF $f_X(x)$ — probability as density (continuous)

The picture. A smooth curve. The shaded area under it between two points is the probability of landing in that stretch. Total area under the whole curve $=1$.

See Probability Distributions for the zoo of PMFs and PDFs.

9. The integral $\int$ — "add up a smooth area"

• $-\infty$ (negative infinity) means "start from the far, far left, before any value is possible."
• $t$ is a dummy variable — a placeholder we slide from $-\infty$ up to $x$. We use $t$ (not $x$) so the moving slider isn't confused with the fixed endpoint $x$.

10. Limits and continuity — the edges of the CDF

The picture. Walk the marker rightward. You have now passed every possible outcome, so you have swept up all the probability — the tally reads $1$. Walk it leftward past everything, and you've passed no outcomes — the tally reads $0$. Both edges matter: a CDF must start at $0$ on the far left and finish at $1$ on the far right.

11. The inverse CDF $F_X^{-1}$ — running the question backwards

The picture. On the CDF graph: pick a height $p$ on the vertical axis, slide right until you first touch the curve, then drop straight down to the horizontal axis — that landing point is $F_X^{-1}(p)$. The "first touch" is what the $\inf$ guarantees when the curve is flat or jumps.

• $F_X^{-1}(0.5)$ = the median (half the probability below it).
• $F_X^{-1}(0.95)$ = the 95th percentile.

This feeds Quantile Functions and Inverse Transform Sampling.

How these feed the topic

Downstream, the CDF powers Empirical Distribution, the Kolmogorov-Smirnov Test, Copulas, Survival Analysis, and Expected Value and Variance.

Equipment checklist

Test yourself — cover the right side and answer out loud.