Foundations — Discrete random variables — PMF, CDF

This page assumes nothing. Before you read the parent note, we build every symbol it quietly uses. Read top to bottom — each block is a brick the next one stands on. We introduce the PMF symbol $p_X$ and CDF symbol $F_X$ formally in §7–§9; until then we only name them in words.

1. The random experiment and its outcomes

Picture it. Roll one die. The outcomes are the six faces. Each face is an $\omega$.

Why we need it. Probability starts from "what can happen?" Before we can attach numbers or chances, we must have a name for "one thing that happened." That name is $\omega$.

2. The sample space $\Omega$

Picture it. A box. Inside the box sit all six dice faces. The box is $\Omega$; each dot inside is an $\omega$.

The symbol $\in$ is read "is an element of" / "is inside." So $3\in \{1,2,3\}$ is true.

3. Sets, events, and the tools $\{\dots \}$, $\cup$, disjoint

• $\{X=x\}$ means "the set of all outcomes $\omega$ for which the number $X$ gives equals $x$." It is therefore an event — a subset of $\Omega$.
• Disjoint sets share no members — like two boxes with nothing in common. Picture two circles that do not overlap.
• $\cup$ read "union" glues sets together into one bigger set: $\{1,2\}\cup \{3\}=\{1,2,3\}$. Picture pouring two boxes into one.

Why the topic needs this. The parent claims $\{X=2\}$ and $\{X=5\}$ are disjoint (a number can't be both 2 and 5), and their union over all values is the whole $\Omega$. That single sentence is what forces the masses to sum to 1 — so you must see the picture behind "event," $\cup$, and "disjoint."

4. Probability $P(\cdot )$ and its three rules

The parent "derives" the PMF properties from three facts (the Probability Axioms). In plain words:

1. Never negative: $P(A)\ge 0$. Chance can't be less than nothing.
2. Total is one: $P(\Omega )=1$. Something in the box definitely happens.
3. Add disjoint pieces: if events don't overlap, the chance of "any of them" is the sum of their chances. This is countable additivity — it works even for an infinite list of pieces.

5. Numbers we map into: $\mathbb{R}$ and functions $X:\Omega \to \mathbb{R}$

Picture it. An arrow leaving each dot in the box $\Omega$ and landing on a spot on the number line $\mathbb{R}$. That bundle of arrows is the random variable $X$.

Why the topic needs it. Raw outcomes ("die shows the face with six pips") are awkward to compute with. $X$ translates each outcome into a number so we can do arithmetic, sum, and average.

6. Countable — the word that makes it "discrete"

7. The PMF symbol $p_X$ and the summation $\sum$

Now we can name the first bookkeeping tool.

• $\sum _{\text{all }x}{p}_X(x)$ means "add $p_X(x)$ over every value $x$ that $X$ can take."
• $\sum _{x_i\le x}{p}_X(x_i)$ means "add only the masses whose value $x_i$ is at most $x$."

Why the topic needs it. The normalization rule ($\sum p_X=1$) and the running total we build in §8–§9 are both written with $\sum$. If $\sum$ is opaque, the central formulas are unreadable — so we pin it down here.

8. The CDF symbol $F_X$ and the $x_i^{-}$ trick

Why the topic needs it. For discrete RVs a single point carries real chance, so whether you include or exclude an endpoint changes the answer. The formula $p_X(x_i)=F_X(x_i)-F_X(x_i^{-})$ literally measures the height of the jump at $x_i$ — the mass sitting exactly there. Picture standing on a stair: your height ($F_X(x_i)$) minus the height a hair to your left ($F_X(x_i^{-})$) equals the step's rise ($p_X(x_i)$).

9. Putting the vocabulary together (this is the parent, in miniature)

Now every symbol in $p_X(x)=P(X=x)$ and $F_X(x)={\sum }_{x_i\le x}{p}_X(x_i)$ has a meaning:

Everything the parent does — masses, staircases, interval formulas — is these bricks stacked.

Prerequisite map

Equipment checklist

Read the question, answer aloud, then reveal. If any trips you, reread that section.

Connections

• Discrete random variables — PMF, CDF — the parent this page prepares you for.
• Probability Axioms — the three rules of $P(\cdot )$ used above.
• Continuous random variables — PDF, CDF — what "countable" rules out: there we integrate.
• Conditional Probability — reuses the same $\Omega$, events, and $P(\cdot )$ vocabulary.