Foundations — Random variables (discrete and continuous)

Before you can read the parent note, you need a small toolbox. Below is every symbol and idea it uses, built from absolutely nothing, each one earning its place before the next arrives.

0. What is a "chance experiment"?

Picture a physical action whose result you cannot predict for sure: flipping a coin, rolling a die, measuring the temperature outside. Each possible result is an outcome.

Look at the figure: the left bag $\Omega$ holds six die faces. That bag is not numbers yet — it holds physical results. Turning those tickets into numbers is the whole job of the next symbol.

The deeper machinery of $\Omega$ (events, how we assign chances to them) lives in Probability Space; here we only need the bag-of-tickets picture.

1. The symbol $P(\cdot )$ — "the chance of"

Picture it as how big a slice of a pie that something owns. The whole pie is size $1$ (it represents "some outcome definitely happens"). A slice can't be negative and can't be bigger than the whole pie — that's why $0\le P\le 1$.

2. The random variable $X$ — the number-making machine

Now we connect the bag of outcomes to numbers.

Look at the arrows in the figure: each die-face ticket on the left is sent by the machine $X$ to a number on the right. The face ⚂ goes to $3$. The machine is fixed; the input is random, so the output is random too. That is exactly why we call $X$ a random variable.

3. "Countable" vs "an interval" — the discrete/continuous split

The parent splits every RV into two families. The dividing word is countable.

Look at the two number lines in the figure. Top: separated dots you can point at one at a time — that is a discrete RV. Bottom: a solid filled bar with no gaps — that is a continuous RV. This picture is the reason discrete RVs use sums ($\sum$) and continuous ones use integrals ($\int$): you add up dots, but you sweep across a bar.

4. The summation symbol $\sum$ — "add these all up"

Picture a checkout receipt: each line is one item's price $g(x)$; $\sum$ is the total at the bottom.

Why this tool and not another? Discrete outputs are separate dots (Section 3). To collect their chances you add them one dot at a time — which is precisely what $\sum$ does. No other operation fits separated points.

5. The integral $\int$ — "add up something spread continuously"

For continuous RVs there are no dots to add; chance is smeared along a bar. We need a tool that adds up a smear.

Look at the shaded region in the figure: each thin vertical strip has height $f(x)$ and tiny width $dx$, so its area is $f(x)dx$ — a sliver of chance. The integral glues all slivers from $a$ to $b$ into one total area = one probability.

Why this tool and not $\sum$? You cannot list the points of a solid bar (Section 3), so summing dot-by-dot is impossible. The integral is the only way to total a continuous smear.

6. Weighting: what "expected value" secretly is

The parent's headline result is the expected value (a long-run average). The idea underneath is weighted average.

Here the weights are the chances $P(X=x)$ (they already sum to $1$ by Section 4). So $E[X]={\sum }_{x}xP(X=x)$ is nothing exotic: each output value pulled by its own chance. Big-chance values pull the average harder — exactly like heavier weights tilting a see-saw further. The full treatment lives in Expectation and Variance.

7. The stray symbols you'll meet

A quick round-up so nothing on the parent page is a surprise.

How these foundations feed the topic

Once these blocks are solid, the parent note's PMFs, PDFs, and $E[X]$ formulas are just these tools snapped together. The named distributions branch off into Bernoulli and Categorical Distributions and the Gaussian Distribution, and the averaging tool grows into Expectation and Variance.

Equipment checklist

Say each answer aloud before revealing. If any stumps you, reread its section.