# Discrete distributions

## Binomial Distribution

binary outcomes (e.g. true or false, head or tails, hit or miss). p and (1-p)

e.g. flipping coin 5 times, P(exactly 2 heads). = 5C2

Total outcomes is 2^5. answer is 5/32

discrete values approach normal distribution.

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

where n is the number of trials

k is the number of successful trials

p is the probability of a successful trial

$$E(X) = np$$

$$VAR(X) = np(1-p)$$

## Poisson Distribution

e.g. X = cars going past my window in the last hour.

Assume hour is random.. no rush hour

previous hour is unrelated. independent.

$$E(X) = \lambda$$

λ cars per hour.

Derivable from binomial. let E(X) = λ = np

n distinct trials (e.g. every millisecond)

p probability of that trial

k required number (e.g. 3 cars going past)

binomial equation with lim n->infinity

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

$$VAR(X) = \lambda$$