PMF

“Probability Mass Function”

Denoted $f x (x)$

F x (x) = - \infty \sum x P r (x = k)

Discrete Models

“k successes in n trials”

P r (X = k) = (n k) p^{k} (1 - p)^{n - k}

“Number of k failures before first success”

P r (Y = k) = (1 - p)^{k} p

P r (Y \geq k) = (1 - p)^{k}

“Number of k failures before r successes”

P r (X = k) = (k + r - 1 k) (1 - p)^{k} p^{r}

“Number of k success items drawn when n draws are made from bin of A success items and B failure items”

P r (X = k) = \frac{( A k ) ( B n - k )}{( A + B n )}

“Number of k successes in a given period of the average number of successes in that period is $λ$ “

P r (X = k) = \frac{λ ^{k} \cdot e ^{- λ}}{k !}

P r (X \geq 2) = 1 - P r (X = 0) - P r (X = 1)

Accumulation of probability

F x (x) = P r (X \leq x)

Denoted $f x (x)$

Like PMF but for continuous probability

Any single value has probability zero:

f x (n) = 0

Any PDF must sum to 1 over $R$ :

\int_{- \infty}^{\infty} f x (x) d x = 1

Integrating a PDF from $- \infty$ to X yields the CDF:

F x (x) = \int_{- \infty}^{x} f x (y) d y