How to read the math

Math notation is a language, not a test

Nothing in this curriculum requires you to be "good at math." It requires you to read math — and reading is a skill you learn once.

Math notation exists for one reason: compression. "The chance that tomorrow's return lands between minus one percent and plus one percent" is 14 words. The same idea in symbols is 10 characters. Once you can read the symbols, the short version becomes easier than the sentence, not harder.

The rule for this entire curriculum: every formula can be read aloud as an English sentence, and the first time a formula appears, we write that sentence out. If you ever hit a formula you cannot read aloud, that is a bug in the lesson — not a gap in you.

Letters stand for unknown numbers

When you see a letter like $X$ or $r$, it is a name for a number whose value we don't know yet.

• $X$ might be "tomorrow's closing price of SPY."
• $r$ might be "next month's return of my portfolio."

The number exists — we just haven't observed it. Naming it lets us talk about it before we know it, the way "the winner of next year's World Cup" names a team before the tournament is played.

By convention, capital letters near the end of the alphabet ($X$, $Y$, $Z$) are used for uncertain quantities. Lowercase letters ($x$, $a$, $b$) usually stand for specific, ordinary numbers — like 5, or 0.02.

Reading P( ... )

$P(\text{something})$ reads: "the probability that something happens." It is always a number between 0 and 1.

• $P(X = 5)$ — "the probability that $X$ turns out to equal 5."
• $P(X > 0)$ — "the probability that $X$ is greater than zero." If $X$ is tomorrow's return, this is the chance tomorrow is an up day.

In words: $P$ is a machine. You put a statement inside the parentheses, and it tells you how likely that statement is to be true.

The comparison symbols

A chained expression describes a range:

$a \le X \le b$

In words: "$X$ lands somewhere between $a$ and $b$, endpoints included." So $P(-1 \le X \le 1)$ reads: "the probability that $X$ lands between minus 1 and plus 1."

A memory hook for $\le$ vs $<$: the extra line under the symbol is the "or equal" part.

Subscripts are item numbers in a list

$X_1, X_2, X_3, \ldots, X_n$

In words: "a list of $n$ quantities, numbered 1 through $n$." If $X_1$ is Monday's return and $X_2$ is Tuesday's, then $X_{47}$ is just "the 47th day's return." The little number identifies which one — it never multiplies or changes anything.

The letter $n$ is the traditional name for "how many items there are." With 252 trading days of data, $n = 252$.

The big sigma means "add them all up"

$\sum_{i=1}^{n} X_i$

In words: "add up all the $X$ values, from item 1 through item $n$." That's the whole meaning. The $\sum$ (Greek capital sigma, chosen because Sum starts with S) is a loop: start the counter $i$ at 1, go up to $n$, and add each $X_i$ to a running total.

Concrete example — three daily returns of 1%, then −2%, then 0.5%:

$\sum_{i=1}^{3} X_i = 0.01 + (-0.02) + 0.005 = -0.005$

As a programmer would write it: total = 0; for i in 1..n: total += X[i]. Every $\sum$ you will ever see is that loop.

Greek letters you will meet soon

Greek letters are just more names — used by convention for particular roles:

• $\mu$ (mu, sounds like "mew") — the average, or center, of a quantity.
• $\sigma$ (sigma, lowercase) — the typical spread around that average. Big $\sigma$ means the value swings wildly; small $\sigma$ means it stays close to $\mu$.
• $\beta$ (beta) — you'll meet it in finance as "how strongly something moves with the market."

You never need to write Greek by hand. You only need to recognize each letter and remember which role it plays.

One more you'll see: E[X]

$\mathbb{E}[X]$ reads: "the expected value of $X$" — the average value $X$ would take over many, many repeats. If a coin flip pays $1 for heads and$0 for tails, the expected value is 0.50: not a value you ever receive on a single flip, but the long-run average per flip. A full lesson (A1-04) builds this idea carefully; for now you only need the pronunciation.

How to read any formula

1. Read left to right, out loud, translating each symbol into words.
2. Do not skim past a formula. A formula is the most information-dense sentence in the lesson — skimming it is skipping the point.
3. If a symbol is unfamiliar, stop. Every symbol in this curriculum is introduced before it is used; check back one lesson or revisit this one.
4. Then check the "In words:" line that follows — it should match the sentence you produced.

Summarize before the next lesson

In two or three of your own sentences: what does $P(\,\cdot\,)$ do, and what kind of thing goes inside its parentheses? Saying it in your own words now will make the next lesson — where we make "probability" precise — feel familiar instead of foreign.