| 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}\, $ | ||
| $ \sin\left(\omega _0 n\right) u[n] \ $ | $ \frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right) $ | ||
