Probability Formulas

Probability Formulas
Properties of Probability Functions
The complement of an event A (i.e. the event A not occurring) $\,P(A^c) = 1 - P(A)\,$
The intersection of two independent events A and B $\,P(A \mbox{ and }B) = P(A \cap B) = P(A) P(B)\,$
The union of two events A and B (i.e. either A or B occurring) $\,P(A \mbox{ or } B) = P(A) + P(B) - P(A \mbox{ and } B)\,$
The union of two mutually exclusive events A and B $\,P(A \mbox{ or } B) = P(A \cup B)= P(A) + P(B)\,$
Event A occurs given that event B has occurred $\,P(A \mid B) = \frac{P(A \cap B)}{P(B)}\,$
Total Probability Law $\,P(B) = P(B|A_1)P(A_1) + \dots + P(B|A_n)P(A_n)\,$

$\mbox{ where } \{A_1,\dots,A_n\} \mbox{ is a partition of sample space } S, B \mbox{ is an event }.$

Bayes Theorem $\,P(A_j|B) = \frac{P(B|A_j)P(A_j)}{\sum_{i=1}^{n}P(B|A_i)P(A_i)},\ \{A_i\} \mbox{ and } B \mbox{ are as above }.$
Expectation and Variance of Random Variables
Binomial random variable with parameters n and p $\,E[X] = np,\ \ Var(X) = np(1-p)\,$
Poisson random variable with parameter $\lambda$ $\,E[X] = \lambda,\ \ Var(X) = \lambda\,$
Geometric random variable with parameter p $\,E[X] = \frac{1}{p},\ \ Var(X) = \frac{1-p}{p^2}\,$
Uniform random variable over (a,b) $\,E[X] = \frac{a+b}{2},\ \ Var(X) = \frac{(b-a)^2}{12}\,$
Gaussian random variable with parameter $\mu \mbox{ and } \sigma^2$ $\,E[X] = \mu,\ \ Var(X) = \sigma^2\,$
Exponential random variable with parameter $\lambda$ $\,E[X] = \frac{1}{\lambda},\ \ Var(X) = \frac{1}{\lambda^2}\,$

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