# How to read math notation

There are some things I never learned (or forgot) about math, starting very simple, like how to read notation.

`P(H)`

```
P = probability
H = hypothesis
P(H) = probability that the hypothesis is true
```

In the case of Bayes Theorem (and maybe others?), `P(H)`

is called the "Prior"

```
E = evidence
P(H|E) = probability that the hypothesis is true GIVEN THAT the evidence is true
```

- Probability of getting a result from
`POP(1)`

given the relative size and percent of`POP(1)`

vs`POP(2)`

- The vertical bar says "GIVEN THAT". It LIMITS the view
- Also called the LIKELIHOOD

## $$\propto$$

Something like this:

$$P(\theta | \Upsilon) \propto P(\Upsilon | \theta)P(\theta)$$

Reads like:

- Probability of theta given upsilon **is proportional to** the probability of upsilon given theta times the probability of theta.

Intersect symbol

- $$\cap$$

- $$X \cap Y$$

- Reads: All the things that are in set X *and* also in set Y

Sigma

$$\sigma$$

Variance. This is the symbol for variance:

$$\sigma^2$$

"Sample mean"

$$\bar{x}$$

- "x with a bar over it"

"Sample variance"

$$s^2_{n-1}$$

- In math notation

- $$\frac{ \sum_{i=1}^{n} (x_i - \bar{x})^2 }{n-1}$$

- Using "n" minus 1 is a better estimate than using "n"

The "hat" latex symbol is

- $$\hat{y}$$

- Used to indicate a regression line

"M" or mu (myu) designates a population mean

$$\mu$$

- This population mean can be used in a subsequent mathematical formula

Union symbol

- $$\cup$$

- $$X \cup Y$$

- Reads: All the elements that are in set X *or* in set Y

Set minus

- $$A \setminus B$$

- Reads: All the things that are in set A that are not in set B

Complement

- If `U`

= the universe, `A`

= a set within the universe, then

- `A'`

is everything in the `U`

that is not in `A`

, or the complement of `A`

, or

- $$A' = U - A$$

- $$A' = U \setminus A$$

The universal set of Integers math notation

- $$\mathbb{Z}$$

- Other "blackboard bold" notation

- Real numbers = $$\mathbb{R}$$

- Rational numbers = $$\mathbb{Q}$$

Membership math notation

- $$-5 \in \mathbb{Z}$$

- Reads: negative five is in the set of integers

- $$X \notin \mathbb{Z}$$

- Reads: the letter X is not in the set of integers

Subset/superset math notation

- Given two sets:

- `A = {1, 3, 5, 7, 18}`

- `B = {1, 7, 18}`

- Then:

- Subset: every element of set `B`

is also in set `A`

: $$B \subseteq A$$

- Superset: Set `A`

contains every element of set `B`

: $$A \supseteq B$$

"Given" symbol

- `P(A | B)`

the `|`

pipe means "given"

- You would read this as the probability of A *given* that B has already happened

Notation similarities

- $$E(X) = \mu_X$$

- The expected value of a random variable X equals the mean of X

- $$Var(X) = E((X - \mu_X)^2) = \sigma_X^2$$

- The variability of random variable X is equal to the expected value of the squared differences between our random variable X and its mean, which is also sigma squared for the random variable X

##### Source:

- Me