Random Variable Reference

Discrete Random Variables

Bernoulli Random Variable

PMF graph:
Notation:	$X \sim Bern (p)$
Description:	A boolean variable that is 1 with probability $p$
Parameters:	$p$ , the probability that $X = 1$ .
Support:	$x$ is either 0 or 1
PMF equation:	$P (X = x) = {\begin{cases} p & if x = 1 \\ 1 - p & if x = 0 \end{cases}$
Expectation:	$E [X] = p$
Variance:	$Var (X) = p (1 - p)$

Parameter

p

Binomial Random Variable

PMF graph:
Notation:	$X \sim Bin (n, p)$
Description:	Number of "successes" in $n$ identical, independent experiments each with probability of success $p$ .
Parameters:	$n \in {0, 1, \dots}$ , the number of experiments. $p \in [0, 1]$ , the probability that a single experiment gives a "success".
Support:	$x \in {0, 1, \dots, n}$
PMF equation:	$P (X = x) = (\binom{n}{x}) p^{x} (1 - p)^{n - x}$
Expectation:	$E [X] = n \cdot p$
Variance:	$Var (X) = n \cdot p \cdot (1 - p)$

Parameter

n

Parameter

p

Poisson Random Variable

PMF graph:
Notation:	$X \sim Poi (λ)$
Description:	Number of events in a fixed time frame if (a) the events occur with a constant mean rate and (b) they occur independently of time since last event.
Parameters:	$λ \in {0, 1, \dots}$ , the constant average rate.
Support:	$x \in {0, 1, \dots}$
PMF equation:	$P (X = x) = \frac{λ^{x} e^{- λ}}{x!}$
Expectation:	$E [X] = λ$
Variance:	$Var (X) = λ$

Parameter

λ

Geometric Random Variable

PMF graph:
Notation:	$X \sim Geo (p)$
Description:	Number of experiments until a success. Assumes independent experiments each with probability of success $p$ .
Parameters:	$p \in [0, 1]$ , the probability that a single experiment gives a "success".
Support:	$x \in {1, \dots, \infty}$
PMF equation:	$P (X = x) = (1 - p)^{x - 1} p$
Expectation:	$E [X] = \frac{1}{p}$
Variance:	$Var (X) = \frac{1 - p}{p^{2}}$

Parameter

p

Negative Binomial Random Variable

PMF graph:
Notation:	$X \sim NegBin (r, p)$
Description:	Number of experiments until $r$ successes. Assumes each experiment is independent with probability of success $p$ .
Parameters:	$r > 0$ , the number of success we are waiting for. $p \in [0, 1]$ , the probability that a single experiment gives a "success".
Support:	$x \in {r, \dots, \infty}$
PMF equation:	$P (X = x) = (\binom{x - 1}{r - 1}) p^{r} (1 - p)^{x - r}$
Expectation:	$E [X] = \frac{r}{p}$
Variance:	$Var (X) = \frac{r \cdot (1 - p)}{p^{2}}$

Parameter

r

Parameter

p

Uniform Random Variable

PDF graph:
Notation:	$X \sim Uni (α, β)$
Description:	A continuous random variable that takes on values, with equal likelihood, between $α$ and $β$
Parameters:	$α \in R$ , the minimum value of the variable. $β \in R$ , $β > α$ , the maximum value of the variable.
Support:	$x \in [α, β]$
PDF equation:	$f (x) = {\begin{cases} \frac{1}{β - α} & for x \in [α, β] \\ 0 & else \end{cases}$
CDF equation:	$F (x) = {\begin{cases} \frac{x - α}{β - α} & for x \in [α, β] \\ 0 & for x < α \\ 1 & for x > β \end{cases}$
Expectation:	$E [X] = \frac{1}{2} (α + β)$
Variance:	$Var (X) = \frac{1}{12} (β - α)^{2}$

Parameter

α

Parameter

β

Exponential Random Variable

PDF graph:
Notation:	$X \sim Exp (λ)$
Description:	Time until next events if (a) the events occur with a constant mean rate and (b) they occur independently of time since last event.
Parameters:	$λ \in {0, 1, \dots}$ , the constant average rate.
Support:	$x \in R^{+}$
PDF equation:	$f (x) = λ e^{- λ x}$
CDF equation:	$F (x) = 1 - e^{- λ x}$
Expectation:	$E [X] = 1 / λ$
Variance:	$Var (X) = 1 / λ^{2}$

Parameter

λ

Normal (aka Gaussian) Random Variable

PDF graph:
Notation:	$X \sim N (μ, σ^{2})$
Description:	A common, naturally occuring distribution.
Parameters:	$μ \in R$ , the mean. $σ^{2} \in R$ , the variance.
Support:	$x \in R$
PDF equation:	$f (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{1}{2} (\frac{x - μ}{σ})^{2}}$
CDF equation:	$\begin{aligned} F (x) & = ϕ (\frac{x - μ}{σ}) & Where ϕ is the CDF of the standard normal \end{aligned}$
Expectation:	$E [X] = μ$
Variance:	$Var (X) = σ^{2}$

Parameter

μ

Parameter

σ

Beta Random Variable

PDF graph:
Notation:	$X \sim Beta (a, b)$
Description:	A belief distribution over the value of a probability $p$ from a Binomial distribution after observing $a + 1$ successes and $b + 1$ fails.
Parameters:	$a \in {1, \dots \infty}$ , the number successes + 1 $b \in {1, \dots \infty}$ , the number of fails + 1
Support:	$x \in [0, 1]$
PDF equation:	$f (x) = B \cdot x^{a - 1} \cdot (1 - x)^{b - 1}$
CDF equation:	No closed form
Expectation:	$E [X] = \frac{a}{a + b}$
Variance:	$Var (X) = \frac{a b}{(a + b)^{2} (a + b + 1)}$

Parameter

a

Parameter

b