How to read math notation

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

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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

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\(\propto\)

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.

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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