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113 Math
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113.010 Math - How to read math notation

# 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:

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

- 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

• Me