Statistics & Probability Formulas

$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i$

mean

$\sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2$

population variance

$s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2$

sample variance

$\sigma=\sqrt{\sigma^2}$

population standard deviation

$s=\sqrt{s^2}$

sample standard deviation

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

$P(A\cap B)=P(A)\cdot P(B)$

if $A$ and $B$ are independent

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

conditional probability — $P(B) \neq 0$

$P(A^c)=1-P(A)$

$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$

Bayes' theorem

$P(n,r)=\frac{n!}{(n-r)!}$

permutations of $r$ items from $n$

$C(n,r)=\binom{n}{r}=\frac{n!}{r!(n-r)!}$

combinations of $r$ items from $n$

$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$

binomial distribution

$f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

normal distribution — probability density function

$z=\frac{x-\mu}{\sigma}$

z-score

$P(X=k)=\frac{\lambda^k e^{-\lambda}}{k!}$

Poisson distribution

$r=\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2\sum(y_i-\bar{y})^2}}$

Pearson correlation coefficient

$b=\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2}$

linear regression slope