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Random Variable Reference


Discrete Random Variables

Bernoulli Random Variable

Notation: $X \sim \Ber(p)$
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: $\p(X=x) = \begin{cases} p && \text{if }x = 1\\ 1-p && \text{if }x = 0 \end{cases}$
PMF (smooth): $\p(X=x) = p^x(1-p)^{1-x}$
Expectation: $\E[X] = p$
Variance: $\var(X) = p (1-p)$
PMF graph:
Parameter $p$:

Binomial Random Variable

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: $$\p(X=x) = {n \choose x}p^x(1-p)^{n-x}$$
Expectation: $\E[X] = n \cdot p$
Variance: $\var(X) = n \cdot p \cdot (1-p)$
PMF graph:
Parameter $n$:
Parameter $p$:

Poisson Random Variable

Notation: $X \sim \Poi(\lambda)$
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: $\lambda \in \{0, 1, \dots\}$, the constant average rate.
Support: $x \in \{0, 1, \dots\}$
PMF equation: $$\p(X=x) = \frac{\lambda^xe^{-\lambda}}{x!}$$
Expectation: $\E[X] = \lambda$
Variance: $\var(X) = \lambda$
PMF graph:
Parameter $\lambda$:

Geometric Random Variable

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: $\var(X) = \frac{1-p}{p^2}$
PMF graph:
Parameter $p$:

Negative Binomial Random Variable

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: $$\p(X=x) = {x - 1 \choose r - 1}p^r(1-p)^{x-r}$$
Expectation: $\E[X] = \frac{r}{p}$
Variance: $\var(X) = \frac{r \cdot (1-p)}{p^2}$
PMF graph:
Parameter $r$:
Parameter $p$:

Continuous Random Variables

Uniform Random Variable

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

Exponential Random Variable

Notation: $X \sim \Exp(\lambda)$
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: $\lambda \in \{0, 1, \dots\}$, the constant average rate.
Support: $x \in \mathbb{R}^+$
PDF equation: $$f(x) = \lambda e^{-\lambda x}$$
CDF equation: $$F(x) = 1 - e^{-\lambda x}$$
Expectation: $\E[X] = 1/\lambda$
Variance: $\var(X) = 1/\lambda^2$
PDF graph:
Parameter $\lambda$:

Normal (aka Gaussian) Random Variable

Notation: $X \sim \N(\mu, \sigma^2)$
Description: A common, naturally occuring distribution.
Parameters: $\mu \in \mathbb{R}$, the mean.
$\sigma^2 \in \mathbb{R}$, the variance.
Support: $x \in \mathbb{R}$
PDF equation: $$f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\Big(\frac{x-\mu}{\sigma}\Big)^2}$$
CDF equation: $$\begin{align} F(x) &= \phi(\frac{x-\mu}{\sigma}) && \text{Where $\phi$ is the CDF of the standard normal} \end{align}$$
Expectation: $\E[X] = \mu$
Variance: $\var(X) = \sigma^2$
PDF graph:
Parameter $\mu$:
Parameter $\sigma$:

Beta Random Variable

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: $a > 0$, the number successes + 1
$b > 0$, 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: $\var(X) = \frac{ab}{(a+b)^2(a+b+1)}$
PDF graph:
Parameter $a$:
Parameter $b$: