Probability Distributions

The distribution of a random variable describes how probability is spread over its possible values. Two random variables with the same distribution behave identically from a probabilistic point of view, even if they come from different experiments. A crit indicator on a d20 and a biased coin that lands heads 5% of the time have the same distribution.

Most distributions used in practice belong to a few families, each indexed by a small number of parameters. We write $X \sim \text{Normal}(\mu, \sigma^2)$ to say that $X$ follows the normal distribution with parameters $\mu$ and $\sigma^2$.

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This chapter covers

• probability mass functions for discrete random variables and probability density functions for continuous ones,
• the discrete families Bernoulli, binomial, categorical and Poisson,
• the continuous families uniform, normal, beta and Student's t,
• the multivariate normal distribution, and
• a summary table of where each family appears in the ML and RL notes.

Probability Mass Functions

A discrete random variable $X$ is described by its probability mass function (PMF), $p(x) = \Pr(X = x)$. A function is a valid PMF exactly when its values are non-negative and add up to one over all possible values,

$$p(x) \ge 0 \quad \text{for all } x, \qquad \sum_{x} p(x) = 1.$$

The probability of any event is the sum of the PMF over the values in it, $\Pr(X \in A) = \sum_{x \in A} p(x)$, and the CDF from chapter 1 is the running sum $F(x) = \sum_{x' \le x} p(x')$.

Probability Density Functions

A continuous random variable $X$ is described by a probability density function (PDF) $f$, whose area over an interval gives the probability that $X$ falls in it,

$$\Pr(a \le X \le b) = \int_a^b f(x) \, dx.$$

A function is a valid PDF exactly when it is non-negative and its total area is one,

$$f(x) \ge 0 \quad \text{for all } x, \qquad \int_{-\infty}^{\infty} f(x) \, dx = 1.$$

The density is the derivative of the CDF, $f(x) = F'(x)$, and for a small width $\Delta x$ the probability of landing near $x$ is approximately $f(x) \, \Delta x$. A density is therefore not a probability, and it can be larger than one. The uniform distribution on $[0, 0.5]$ has density $2$ everywhere on that interval, and its total area is still $0.5 \times 2 = 1$.

Discrete Distributions

Bernoulli

The Bernoulli distribution describes a single trial with two outcomes, success ($X = 1$) with probability $p$ and failure ($X = 0$) with probability $1 - p$. Its PMF is

$$\Pr(X = k) = p^k (1-p)^{1-k}, \qquad k \in \{0, 1\}.$$

The crit indicator $C$ from chapter 1 is $\text{Bernoulli}(0.05)$. In machine learning, logistic regression models a binary label as a Bernoulli random variable whose $p$ depends on the input.

Probability of success p0.5

Binomial

The binomial distribution counts the successes in $n$ independent Bernoulli trials with the same success probability $p$. Independent means that the result of one trial doesn't change the probabilities of the others. There are $\binom{n}{k}$ ways to place $k$ successes among $n$ trials, each with probability $p^k (1-p)^{n-k}$, so

$$\Pr(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \qquad k = 0, 1, \dots, n.$$

The number of critical hits in 20 attack rolls is $\text{Binomial}(20, 0.05)$. The probability of at least one crit is $1 - \Pr(X = 0) = 1 - 0.95^{20} \approx 0.64$, so a full evening of combat without a single crit isn't that unusual.

Number of trials n20Probability of success p0.05

Categorical

The categorical distribution generalizes the Bernoulli to $K$ outcomes. The random variable $X$ takes a value in $\{1, \dots, K\}$, and a vector of probabilities $\boldsymbol{\pi} = (\pi_1, \dots, \pi_K)$ gives

$$\Pr(X = k) = \pi_k, \qquad \pi_k \ge 0, \quad \sum_{k=1}^K \pi_k = 1.$$

A fair d20 is categorical with $\pi_k = 1/20$ for every face. Softmax regression outputs a categorical distribution over classes, and in RL a stochastic policy $\pi(a \mid s)$ over a finite set of actions is a categorical distribution for each state (see Markov Decision Processes). Counting how often each outcome occurs in $n$ categorical trials gives the multinomial distribution, which relates to the categorical the way the binomial relates to the Bernoulli.

Poisson

The Poisson distribution counts events in a fixed interval of time or space, when events occur independently of each other at a constant average rate $\lambda$. Its PMF is

$$\Pr(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \qquad k = 0, 1, 2, \dots$$

The number of random encounters during a day of travel, or of emails you receive in an hour, are classic examples. The Poisson is also the limit of the binomial for rare events. If $n$ is large and $p = \lambda / n$ is small, $\text{Binomial}(n, p)$ is close to $\text{Poisson}(\lambda)$. Poisson regression uses it to model counts.

Rate λ4

Continuous Distributions

Uniform

The uniform distribution on $[a, b]$ spreads probability evenly over the interval,

$$f(x) = \begin{cases} \frac{1}{b-a} & \text{if } a \leq x \leq b, \\ 0 & \text{otherwise.} \end{cases}$$

Random number generators produce $\text{Uniform}(0, 1)$ values, and samplers for every other distribution, including the ones in these interactives, start from them. Shrink the interval below a width of one to see the density rise above one.

Lower bound a0Upper bound b1

Normal

The normal (or Gaussian) distribution is the bell curve,

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right),$$

where $\mu$ sets the center and $\sigma > 0$ sets the spread. About 68% of the probability lies within one $\sigma$ of $\mu$, 95% within two and 99.7% within three. The standard normal has $\mu = 0$ and $\sigma = 1$, and any normal random variable can be standardized to it with $Z = (X - \mu)/\sigma$.

The normal shows up so often because averages of many independent quantities are approximately normal, which is the central limit theorem. The ML notes use it to model the noise in linear regression, and the RL notes use it for policies over continuous actions in policy gradient methods.

Mean μ0Standard deviation σ1

Beta

The beta distribution lives on $[0, 1]$, which makes it a natural way to describe uncertainty about an unknown probability. With shape parameters $\alpha, \beta > 0$,

$$f(x) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)}, \qquad 0 \le x \le 1,$$

where $B(\alpha, \beta) = \Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha + \beta)$ makes the area equal to one. With $\alpha = \beta = 1$ it is the uniform distribution. The mean is $\alpha / (\alpha + \beta)$, and the larger $\alpha + \beta$ is, the more concentrated the distribution becomes.

If you suspect that a d20 is loaded but don't know by how much, a beta distribution can describe your belief about its crit chance. A later chapter on Bayesian inference updates that belief with each roll, and the same idea drives Thompson sampling for bandits.

α2β5

Student's t

The Student's t distribution with $\nu$ degrees of freedom has density

$$f(x) = \frac{\Gamma \left(\frac{\nu+1}{2}\right)}{\sqrt{\nu \pi} \, \Gamma \left(\frac{\nu}{2}\right)} \left(1 + \frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}}.$$

It is bell-shaped and centered at zero like the standard normal, but with heavier tails, and it approaches the standard normal as $\nu$ grows. It appears when a sample mean is standardized with the sample's own standard deviation instead of the true one, which is the basis of confidence intervals and t-tests in the planned inference chapters.

Degrees of freedom ν3

Multivariate Normal

Many quantities in ML are vectors, such as the weights of a model or the coordinates of a data point. A random vector $\mathbf{X} \in \mathbb{R}^d$ follows the multivariate normal distribution $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ when its density is

$$f(\mathbf{x}) = \frac{1}{(2\pi)^{d/2} \det(\boldsymbol{\Sigma})^{1/2}} \exp\left(-\frac{1}{2} (\mathbf{x} - \boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu})\right).$$

The mean vector $\boldsymbol{\mu}$ sets the center, and the covariance matrix $\boldsymbol{\Sigma}$, which must be symmetric and positive definite, sets the spread along each direction and how the coordinates move together. Points of equal density form ellipses (ellipsoids in higher dimensions). When $\boldsymbol{\Sigma}$ is diagonal the ellipses line up with the axes and the coordinates are independent, and when $\boldsymbol{\Sigma} = \sigma^2 \mathbf{I}$ they are circles. The probabilistic interpretation of linear regression uses this distribution for the vector of targets.

Summary

The mean and variance columns are defined in the next chapter.

Distribution	Values	Parameters	Mean	Variance	Where it appears
Bernoulli	$\{0, 1\}$	$p$	$p$	$p(1-p)$	Binary labels, logistic regression
Binomial	$\{0, \dots, n\}$	$n, p$	$np$	$np(1-p)$	Counting successes
Categorical	$\{1, \dots, K\}$	$\pi_1, \dots, \pi_K$			Softmax regression, stochastic policies
Poisson	$\{0, 1, 2, \dots\}$	$\lambda$	$\lambda$	$\lambda$	Count data, Poisson regression
Uniform	$[a, b]$	$a, b$	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$	Random number generation
Normal	$\mathbb{R}$	$\mu, \sigma^2$	$\mu$	$\sigma^2$	Regression noise, Gaussian policies, the CLT
Beta	$[0, 1]$	$\alpha, \beta$	$\frac{\alpha}{\alpha+\beta}$	$\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$	Beliefs about probabilities, Thompson sampling
Student's t	$\mathbb{R}$	$\nu$	$0$ for $\nu > 1$	$\frac{\nu}{\nu-2}$ for $\nu > 2$	Confidence intervals, t-tests

The categorical's outcomes are labels rather than quantities, so its mean and variance are usually not used.

1. A player makes 20 attack rolls with a fair d20. Which distribution describes the number of critical hits?

• Bernoulli with $p = 0.05$
• Binomial with $n = 20$ and $p = 0.05$
• Uniform on $[0, 20]$
• Categorical with 20 equally likely outcomes

2. Which statement about a probability density function $f$ is true?

• $f(x)$ can be larger than 1, as long as the total area is 1
• $f(x)$ is the probability that $X = x$
• $f(x)$ must add up to 1 over all $x$
• $f(x)$ can never be larger than 1

3. What happens to Student's t distribution as the degrees of freedom $\nu$ grow?

• It approaches the standard normal
• Its tails get heavier
• Its mean moves away from zero
• It approaches the uniform distribution

What's Next

The summary table lists a mean and a variance for each family. Expectation, Variance, and Covariance defines these quantities and the rules for computing them.