Discrete Distributions

Discrete Uniform Distribution

$f (x) = \frac{1}{b - a + 1}$
$f (x) =$ pdf
- probability that we get x = function

Hypergeometric Distribution

without replacement
$f (x) = \frac{(\binom{r}{x}) (\binom{N - r}{n - x})}{(\binom{N}{n})}$
- $N =$ total, $n =$ sample size, $x =$ target, $r =$ group we choose x from
expectation
- $E [X] = (\frac{r n}{N})$
  - expectation of the value of X in a group of size n

Binomial Distribution

independent trials/with replacement
$f (x) = (\binom{n}{x}) p^{x} (1 - p)^{n - x}$
- n = total, x = number of successes, p = probability of success
expectation
- $n p$

Negative Binomial Distribution

binomial until a set number of successes occur
- independent trials/with replacement
$f (x) = (\binom{x + k - 1}{x}) p^{k} (1 - p)^{x}$
- k = number of successes
- $x =$ number of failures before the kth success
- $k - 1$ as the last success is already set at last spot
expectation
- $E | X | = \frac{k (1 - p)}{p}$

Poisson Distribution

$f (x) = e^{- μ} \frac{μ^{x}}{x!}, x = 0, 1, 2, . . .$
$P(X=x)=\frac{\lambda {#xe} ^{-\lambda}}{x!}$
$μ = E (X)$ , use $E (X)$ formula from
conditions of a poisson
- independence
  - number of occurrences in non-overlapping intervals need to be independent
- individuality
  - events can only have one occurrence at a time
  - for sufficiently short time periods of length $Δ t$ , the probability of 2 or more events occurring in the interval is close to zero i.e. events occur singly not in clusters. More precisely, as $Δ t \to 0$ , the probability of two or more events in the interval of length $Δ t$ must go to zero faster than $Δ t \to 0$
- uniformity
  - events occur at a uniform or homogenous rate $λ$ over time so that the probability of one occurrence in an interval $(\vec{t}, t + Δ t)$ is approximately $λ Δ t$ for any value of t