21  Common Distributions

Discrete distributions

Bernoulli distribution

A random variable X \in \{0, 1\} follows the Bernoulli distribution

X \sim \mathrm{Ber}(p) \quad p \in [0, 1],

if X takes value 1 (success) with probability p and 0 (failure) with probability 1 - p.

\mathbb{P}_{X}(x) = p^{x} (1 - p)^{1 - x}

\mathbb{E}_{X}[x] = p

\mathrm{Var} [x] = p (1 - p)

Binomial distribution

A random variable $X {0, , n} $ follows the binomial distribution

X \sim \mathrm{Bin}(n, p) \quad n \in \mathbb{N} \quad p \in [0, 1],

if X is the sum of the results of (or number of successes in) n independent and identically distributed Bernoulli trials with probability p.

\mathbb{P}_{X}(x) = {n \choose x} p^{x} (1 - p)^{n - x}

\mathbb{E}_{X}[x] = np

\mathrm{Var} [x] = np(1 - p)

Geometric distribution

A random variable X \in \{ 1, 2, \dots\} follows the geometric distribution

X \sim \mathrm{Geo}(p) \quad p \in [0, 1],

if X is the number of independent Bernoulli trials with parameter p up to and including first success.

\mathbb{P}_{X}(x) = p (1 - p)^{x - 1}

\mathbb{E}_{X}[x] = \frac{1}{p}

\mathrm{Var} [x] = \frac{1}{p^2}

Negative binomial distribution

A random variable X \in \{ r, r + 1, \dots \} follows the negative binomial distribution

X \sim \mathrm{NegBio}(r, p) \quad r \in \mathbb{N} \quad p \in [0, 1],

if X is the number of independent Bernoulli trials with parameter p up to and including the r successes.

\mathbb{P}_{X}(x) = {x - 1 \choose r - 1} p^{r} (1 - p)^{x - r}

\mathbb{E}_{X}[x] = \frac{r}{p}

\mathrm{Var} [x] = \frac{r (1 - p)}{(1 - p)^2}

Poisson distribution

A random variable X \in \mathbb{N} follows the Poisson distribution

X \sim \mathrm{Poi}(\lambda) \quad \lambda > 0,

if X is the number of events that occur in one unit of time independently with rate \lambda per unit time.

\mathbb{P}_{X}(x) = e^{-\lambda} \frac{\lambda^{x}}{x!}

\mathbb{E}_{X}[x] = \lambda

\mathrm{Var} [x] = \lambda

Continuos distribution

Uniform distribution

A random variable X \in \mathbb{R} follows the Uniform distribution

X \sim \mathrm{Unif} (a, b) \quad a < b,

if X describes an experiment whose outcomes are equally likely in a range.

\mathbb{P}_{X} (x) = \begin{cases} \begin{aligned} & \frac{ 1 }{ b - a } && \quad x \in [a, b] \\ & 0 && \quad \text{otherwise} \end{aligned} \end{cases}

\mathbb{E}_{X} [X] = \frac{ a + b }{ 2 }

\mathrm{Var} [X] = \frac{ (b - a)^{2} }{ 12 }

F_{X} (x) = \begin{cases} \begin{aligned} & 0 && \quad x < a \\ & \frac{ x - a }{ b - a } && \quad a \leq x \leq b \\ & 1 && \quad x > b \end{aligned} \end{cases}

Exponential distribution

A random variable X \in [0, \infty] follows the Exponential distribution

X \sim \mathrm{Exp} (\lambda) \quad \lambda \in \mathbb{R},

if X is the waiting time until the first occurrence of an event in a Poisson Process with parameter \lambda.

\mathbb{P}_{X} (x) = \begin{cases} \begin{aligned} & \lambda e^{- \lambda x} && \quad x \geq 0 \\ & 0 && \quad \text{otherwise} \end{aligned} \end{cases}

\mathbb{E}_{X} [X] = \frac{ 1 }{ \lambda }

\mathrm{Var} (X) = \frac{ 1 }{ \lambda^{2} }

F_{X} (x) = \begin{cases} \begin{aligned} & 1 - e^{- \lambda x} && \quad x \geq 0 \\ & 0 && \quad \text{otherwise} \end{aligned} \end{cases}

Gaussian (normal) distribution

A random variable X \in \mathbb{R} follows the Gaussian distribution

X \sim \mathcal{N} (\mu, \sigma) \quad \mu \in \mathbb{R}, \sigma \in \mathbb{R},

if X follows the standard bell curve.

\mathbb{P}_{X} (x) = \frac{ 1 }{ \sigma \sqrt{2 \pi} } \exp \left[ - \frac{ (x - \mu)^{2} }{ 2 \sigma^{2} } \right]

\mathbb{E}_{X} [X] = \mu

\mathrm{Var} (X) = \sigma^{2}