Foundations — Expected value, variance, standard deviation — properties

Before you can read a single formula in Expected value, variance, standard deviation — properties, you need to own every symbol it throws at you. Below, each idea is built from nothing, tied to a picture, and justified — then the next idea stands on it.

1. A number line and a point on it

Everything starts with the number line: a horizontal ruler where every position is a number. Small numbers on the left, big numbers on the right, $0$ in the middle.

A single point on that line is one specific outcome — say, "the die showed a $4$", which we mark at position $4$.

We need this because a random variable's job is to land somewhere on this ruler, and both of our summaries (center, spread) are described as locations and distances on it.

2. The symbol $X$ — a random variable

Picture: think of a dart that will land somewhere on the ruler. $X$ is "wherever the dart lands." Roll a die → $X$ is the face. Flip a coin and score heads as $1$ → $X$ is $0$ or $1$.

Why the topic needs it: all the formulas — $\mathbb{E}[X]$, $\mathrm{Var}(X)$ — are questions about this dart's habits. Capital letters like $X,Y$ always mean "a random number." Lowercase $x_i$ means "one specific possible value it could take."

3. Probability $p_i$ — how much mass sits at each value

Picture: stack a lump of clay at each possible value. A taller lump = more likely. The total clay must weigh exactly $1$ — because something always happens.

The symbol ${\sum }_i$ (capital Greek sigma) just means "add up over all the possibilities $i$." Read ${\sum }_i{p}_i$ as "add every lump's weight." That total is $1$ — this fact is used constantly in the parent note to make terms collapse.

For continuous variables (which can be any real number, not just a list), the clay is smeared into a smooth curve $f(x)$ called the density. Then "adding up" becomes an integral $\int$, which is just a sum over infinitely many infinitely-thin slices.

4. Expected value $\mathbb{E}[X]$ and its symbol $\mu$

Picture: put your finger under the ruler and slide it until neither side tips. That finger position is $\mathbb{E}[X]=\mu$.

Why "opposite × mass"? Each lump pulls the balance toward its position $x_i$, with pulling strength equal to its weight $p_i$. Averaging position weighted by mass gives ${\sum }_i{x}_i{p}_i$ — the physics center of mass. That is why the formula in the parent looks the way it does.

5. Distance from the mean, and why we square it

To talk about spread, we ask: how far is a typical outcome from the center $\mu$?

The deviation of a value is $x_i-\mu$: how far right ($+$) or left ($-$) of the balance point it sits.

We remove signs by squaring: $(x_i-\mu {)}^2$ is always $\ge 0$, and it punishes far-away points more than nearby ones (a deviation of $3$ contributes $9$).

6. Variance $\mathrm{Var}(X)$ and standard deviation $\sigma$

Picture: for each lump, measure its distance to the finger, square it, then take the mass-weighted average of those squares. A wide spread → big squared distances → big variance.

Units problem: if $X$ is in metres, squared distances are in m². To get back to plain metres we take a square root.

The symbol $\sigma$ (Greek "sigma", lowercase) always means standard deviation; ${\sigma }^2$ therefore means variance. That is why the parent freely writes both.

7. Functions of $X$, and the symbol $g(X)$

Sometimes we care not about $X$ but about a rule applied to it — call the rule $g$. So $g(X)$ is "take whatever $X$ landed on, then transform it."

• $g(x)=x^2$ → $g(X)=X^2$ (used for variance).
• $g(x)=ax+b$ → a straight-line rescale (used for scale/shift rules).

Why the topic needs it: $\mathbb{E}[X^2]$ in the computational formula is exactly LOTUS with $g(x)=x^2$. Without this shortcut you'd have to rebuild the whole distribution of $X^2$.

8. A second variable $Y$, and how two variables relate

To add randomness (like summing coin flips), we need a second random variable $Y$: another dart on the same ruler.

Full detail lives in Covariance and Correlation; here you only need to recognise the symbol and know that independence makes it vanish — which is why the sum-of-variances rule needs independence.

9. Prerequisite map

Read it top-down: geometry (number line) feeds the random variable and probability mass; those feed the mean; the mean feeds deviations, squares, variance, and finally standard deviation. Functions and a second variable branch off to power LOTUS and the sum rule. These foundations connect onward to Law of Large Numbers and Central Limit Theorem, which describe what happens to means and spreads as you collect many samples.

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