Probability, by hand.

Slide the parameters, shade an interval to read off a probability, then draw samples and watch the histogram settle onto the curve.

The essentials

Random variable

X : \Omega \to \mathbb{R}

A number whose value depends on chance, like the result of a die roll or tomorrow's rainfall.

Discrete vs continuous

\{0, 1, 2, \dots\} \quad \text{vs} \quad [a, b] \subseteq \mathbb{R}

Discrete variables take separate values you can list (0, 1, 2, …). Continuous ones can land anywhere in a range, so any single exact value has probability zero.

PMF (discrete)

p(k) = P(X = k), \quad \sum_k p(k) = 1

The probability mass function gives the probability of each exact value. The bars always add up to 1.

PDF (continuous)

P(a \le X \le b) = \int_a^b f(x)\,dx

The probability density function is not a probability by itself. Probability is the area under it, so f(x) can be bigger than 1.

CDF

F(x) = P(X \le x)

The cumulative distribution function climbs from 0 to 1. The probability of landing between a and b is F(b) − F(a).

Expectation

\mathbb{E}[X] = \sum_k k\,p(k) ;\text{ or }; \int x f(x)\,dx

The long-run average, and the balance point of the distribution. Draw enough samples and their mean settles here.

Variance and standard deviation

\operatorname{Var}(X) = \mathbb{E}[X^2] - \mu^2, \quad \sigma = \sqrt{\operatorname{Var}(X)}

How spread out the values are around the mean. The standard deviation is in the same units as X.

Central Limit Theorem

\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} \mathcal{N}(0, 1)

Average enough independent draws from almost any distribution and the average looks normal. Try the Sample means mode on any distribution page.

Discrete

Countable outcomes: heads or tails, the number of emails, the roll of a die.

Bernoulli

$X \sim \operatorname{Bernoulli}(p)$

A single yes/no trial that succeeds with probability p.

Binomial

$X \sim \operatorname{Bin}(n, p)$

How many successes in n independent yes/no trials.

Geometric

$X \sim \operatorname{Geom}(p)$

How many trials until the first success.

Poisson

$X \sim \operatorname{Pois}(\lambda)$

How many events happen in a fixed window when they arrive at a steady average rate.

Discrete Uniform

$X \sim \mathcal{U}\{a, \dots, b\}$

Every whole number from a to b is equally likely, like a fair die.

Continuous

Any value in a range: heights, waiting times, probabilities themselves.

Uniform

$X \sim \mathcal{U}(a, b)$

Any value between a and b is equally likely.

Normal (Gaussian)

$X \sim \mathcal{N}(\mu, \sigma^2)$

The bell curve. Sums of many small effects end up looking like this.

Exponential

$X \sim \operatorname{Exp}(\lambda)$

The waiting time until the next event in a Poisson process.

Gamma

$X \sim \Gamma(k, \theta)$

The waiting time for k events. A flexible, skewed shape for positive values.

Beta

$X \sim \operatorname{Beta}(\alpha, \beta)$

A distribution over probabilities themselves, living between 0 and 1.

Chi-squared

$X \sim \chi^2(k)$

The sum of k squared standard normals. The workhorse of hypothesis tests.

Student's t

$X \sim t_\nu$

A bell curve with heavier tails, for when σ has to be estimated from a small sample.

Log-normal

$X \sim \operatorname{LogN}(\mu, \sigma^2)$

A variable whose logarithm is normal. Products of many small effects end up like this.