Discrete Distributions

The Atoms of Probability: Bernoulli Trials

Every complex system starts with a simple choice: Yes or No? In probability, this is a Bernoulli Trial. It's a single experiment with exactly two outcomes: Success (1) or Failure (0). We define $p$ as the probability of success.

EExample

Bernoulli Examples

A single coin flip (Heads = Success, $p=0.5$ ).
A single customer deciding whether to buy a product ( $p=0.1$ ).
A single pixel in an image being "hot" or "dead" ( $p=0.0001$ ).

Summary of Discrete Models

From this simple Bernoulli trial, we can build a vast array of discrete distributions depending on what we choose to measure and fix. Over the next few lessons, we'll dive deep into each of the core discrete distributions:

Distribution	Core Question	Mean (E[X])	Variance (Var(X))
Bernoulli	Single trial success/failure?	$p$	$p(1-p)$
Binomial	How many successes in $n$ fixed trials?	$np$	$np(1-p)$
Geometric	How many trials until the 1st success?	$1/p$	$(1-p)/p^2$
Hypergeometric	Successes in $n$ trials (without replacement)?	$n \frac{K}{N}$	$n \frac{K}{N} \frac{N-K}{N} \frac{N-n}{N-1}$
Multinomial	Counts of $k$ distinct categories in $n$ trials?	$np_i$	$np_i(1-p_i)$
Poisson	How many rare events in a fixed time/space?	$\lambda$	$\lambda$

Let's begin by exploring the most fundamental extension of the Bernoulli trial: the Binomial Distribution.

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Random Variables & Expectations

Course Progression

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The Binomial Distribution