Probability and Statistics Formulas

Bernoulli (pmf, expectation, variance, applications)

PMF: $p^{t}\left(1-p\right)^{1-t}$

Expectation: E[x] = p

Variance: Var(x) = p(1-p)

Applications: Modeling a single event outcome (e.g., default/no default, success/failure of a trade).

Binomial (pmf, expectation, variance, applications)

PMF: P(x = k) =

(n choose k) $\cdot p^{k}\cdot\left(1-p\right)^{n-k}$

Variance, Standard Deviation (equations)

Variance: Var(X) = E[X²] - (E[$X$])²

Standard Deviation: $\sigma=\sqrt{Var\left(x\right)}$

Conditional Probability (equation and description )

$$P\left(A\vert B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}$$

The probability of event A occurring given that event B has already occurred

Bayes’ Theorem (equation and description)

$$P\left(A_1\vert B\right)=\frac{P\left(A_1\right)\cdot P\left(B\vert A_1\right)}{P(B)}$$

Relates posterior probability to prior and likelihood (crucial for updating beliefs as new data arrives)