Formula sheet

Counting

$$_{n}P_{r} = \frac{n!}{(n - r)!},\quad\quad\binom{n}{r} = \frac{n!}{r!(n - r)!}.$$

$$\text{Repeated arrangements} = \frac{n!}{n_{1}!n_{2}!\cdots n_{k}!}.$$

Probability

$$P(A \cup B) = P(A) + P(B) - P(A \cap B).$$

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}.$$

$$P(B) = \sum_{i}^{}P\left( B \mid A_{i} \right)P\left( A_{i} \right).$$

$$P\left( A_{j} \mid B \right) = \frac{P\left( B \mid A_{j} \right)P\left( A_{j} \right)}{\sum_{i}^{}P\left( B \mid A_{i} \right)P\left( A_{i} \right)}.$$

Expectation and Variance

$$Var(X) = E\left( X^{2} \right) - \left\lbrack E(X) \right\rbrack^{2}.$$

$$Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y).$$

Distribution Summary

Distribution	Support	Mean	Variance
Bernoulli$(p)$	$0,1$	$p$	$p(1 - p)$
Binomial$(n,p)$	$0,\ldots,n$	$np$	$np(1 - p)$
Geometric$(p)$, trial count	$1,2,\ldots$	$1/p$	$(1 - p)/p^{2}$
Negative binomial$(r,p)$, trial count	$r,r + 1,\ldots$	$r/p$	$r(1 - p)/p^{2}$
Poisson$(\lambda)$	$0,1,\ldots$	$\lambda$	$\lambda$
Hypergeometric$(N,K,n)$	finite integer range	$nK/N$	$np(1 - p)(N - n)/(N - 1)$
Uniform$(a,b)$	$a \leq x \leq b$	$(a + b)/2$	$(b - a)^{2}/12$
Normal$\left( \mu,\sigma^{2} \right)$	$\mathbb{R}$	$\mu$	$\sigma^{2}$
Gamma$(\alpha,\beta)$, rate form	$x > 0$	$\alpha/\beta$	$\alpha/\beta^{2}$
Chi-square$(\nu)$	$x > 0$	$\nu$	$2\nu$