Probability

$Var(X) = E(X - E(X))^2 = E(X^2) - E(X)^2 = M2 - M1$ (moments)
$Cov(X,Y) = E ((X −E(X))(Y −E(Y))) = E(XY)−E(X)E(Y)$
- $Var(X) = Cov(X,X)$
Standard Deviation = $\sqrt{Var(x)}$
$P(A \cup B) = P (A) + P(B) - P(A \cap B)$
Marginal distribution: $P(X = x) = \sum_y P(X = x, Y = y)$
If A and B are independent events: $P(A \vert B) = P(A)$ and $P(A ∩ B) = P(A) . P(B)$
- If A and B are independent: $Cov(A,B) = 0$
A and B are conditionally independent if: $P((A ∩ B)/C) = P(A \vert C) . P(B \vert C)$

Bayes’ Formula: $P(A \vert B) = {P(B \vert A)P(A) \over P(B)}$

\[P(A \vert B) = {P(A \cap B) \over P(B)} \text{ or } P(A ∩ B) = P(A \vert B) . P(B) = P(B \vert A) . P(A)\]

In other words: $\text{posterior} = {\text{likelihood} * \text{prior} \over \text{evidence}}$

Law of Total Probability $P(A) = P(A \vert B1).P(B1) + P(A \vert B2).P(B2) + ... + P(A \vert Bn).P(Bn)$ or, $p(x)=\sum_1^cp(x \vert y_i)P(y_i)$

TODO: