Distributions

Think of these as a class in Python or some other OOP language. If you identify that an RV 'fits' one of these, you can use precalculated/derived formulae for things like the Probability Mass Function, Expected Value $E(X)$, and Variance $Var(X)$.

Don't Overthink it, Sunshine

A 'distribution' doesn't have to be anything fancy and can just be a small list that tells people what the probabilities are for each value ("realization") of your Random Variable. E.g. your shiny variable $X$ can only take on values $\{1, 2, 3\}$ and you assign each value ("realization") probabilities of $\{\frac{2}{9}, \frac{4}{9}, \frac{3}{9}\}$. Bam, done, Distribution.

This applies to joint distributions as well! Let's say $X$ can take on $\{1,2\}$ and $Y$ can take on $\{a, b\}$. You'd do a nice table of $P(X = 1, Y = a)$, $P(X = 2, Y = a)$, $P(X = 1, Y = b)$, and $P(X = 2, Y = b)$. Note that I'm not saying anything about independence or anything. Just a table of values (which may reveal stuff about the relation between $X$ and $Y$).

There are rules of course. But that's all really. A function is nice because you won't have to labor over this list for a lot of values (and if you're laboring over $\Reals$, you'll be laboring for a long time indeed).

The Normal Distribution ♥️

TODO: Write more about this and the CLT.

Here's a lovely derivation that's worth watching.

Discrete Distributions

Bernoulli

Simplest one. If an experiment with probability $p$ succeeds, you get a 1, else 0.

$$X \sim Ber(p)$$

PMF	$E[X]$	$Var(X)$
$P(X = 1) = p$, $P(X = 0) = 1-p$	$p$	$p(1-p)$

Examples: Coin flip results in a heads. Your friend liked Nights in Rodanthe.

Binomial

Extends Bernoulli to $n$ independent trials of your experiment with probability of success $p$

$$X \sim Bin(n,p)$$

PMF	$E[X]$	$Var(X)$
$P(X = k) = {n \choose k} p^k(1-p)^{n-k}$	$np$	$np(1-p)$

Examples: Number of heads in 12 coin flips. Bits being corrupted (think Hamming Codes).

Poisson

$$X \sim Poi(\lambda)$$

Approximates Binomial to cases when probablity $p$ is very small and the number of trials $n$ is very large. It's pretty important and used in queueing theory.

• Think the number of things that happen over a fixed interval of time at a constant average rate.
• Think of events that are rare, independent (of course), and relatively uniformly distributed in time or space: mutations in a stretch of DNA, calls into a call center, insurance claims filed throughout a day, patients arriving at a clinic, rare diseases reported per month, meteors observed in the night sky, bankruptcies observed in a year...

Geometric

$$X \sim Geo(p)$$

Number of independent trials until first success (which has a probability $p$).

Negative Binomial

$$X \sim NegBin(p)$$

This is the number of independent trials you'll need until a fixed number of successes $r$ with probability $p$. Kinda extends the idea of the Geometric.