## Bernoulli Distribution ### Conditions for a Bernoulli

• There is one trial
• There are two possible outcomes, success or failure

### Formulae

P(X = x) = px(1 - p)1-x   x ∈ {0, 1}
P(X = 0) = 1 - p
P(X = 1) = p
E(X) = p
VAR(X) = p(1 - p)

## Binomial Distribution ### Conditions for a Binomial

• The number of trials is fixed
• The trials are independent of each other
• There are two possible outcomes, success or failure
• The probability of success is constant

### Formulae

P(X = x) = nCx px(1 - p)n-x
=
 n| x! (n - x)!
× px(1 - p)n-x
E(X) = np
VAR(X) = np(1 - p)

## Geometric Distribution ### Conditions for a Geometric Distribution

• The trials are independent of each other
• There are two possible outcomes, success or failure
• The probability of success is constant

### Formulae

P(X = x) = p(1 - p)x-1
P(X > x) = (1 - p)x
P(X ≤ x) = 1 - (1 - p)x
E(X) =
 1 p
VAR(X) =
 (1 - p) p2

## Negative Binomial Distribution ### Conditions for a Negative Binomial Distribution

• The trials are independent of each other
• There are two possible outcomes, success or failure
• The probability of success is constant

### Formulae

P(X = x) = x-1Cr-1pr(1 - p)x-r
=
 (x - 1)| (r - 1)! (x - r)!
× pr(1 - p)x-r
E(X) =
 r p
VAR(X) =
 r(1 - p) p2

## Poisson Distribution ### Conditions for a Poisson Distribution

• Each event is independent
• Events occur at a constant average rate
• Events cannot occur at the same time

### Formulae

P(X = x) =
 e-λλx x!
E(X) = λ
VAR(X) = λ

## Uniform Distribution ### Conditions for a Discrete Uniform Distribution

• All outcomes are evenly distributed (difference is one for this application)
• All outcome are equally as likely

### Formulae

P(X = x) =
 1 n
E(X) =
 a + b 2
VAR(X) =
 n2 - 1 12