Common discrete distributions — Bernoulli, Binomial, Poisson, Geometric, Negative Binomial: Foundations

This page assumes nothing. We introduce every symbol the parent note uses, in the order that each one needs the previous. If a symbol appears on the parent page and you are not 100% sure what its little marks mean — it is defined below.

0. Powers and the exponential — notation used everywhere below

Before anything else, two pieces of notation appear on nearly every line of the parent page: powers and the exponential function. We pin them down now so nothing later is a surprise.

The picture. $x^{k}$ is repeated stretching: start at $1$ and stretch by a factor $x$ , $k$ times in a row. In our formulas $q^{k - 1}$ is " $q$ multiplied by itself $k - 1$ times" — the chance of $k - 1$ independent failures in a row.

Why the topic needs it. Multiplying independent probabilities produces powers like $p^{k}$ and $q^{n - k}$ ; the Poisson limit produces $e^{- λ}$ . Powers are the language every distribution is written in.

1. What is a "random variable"? The symbol $X$

The picture. Imagine a box (the experiment). Something random happens inside — a coin flip, a die roll. Out comes a number. $X$ is the label on the box; $k$ is the number that pops out this time.

Figure — Common discrete distributions — Bernoulli, Binomial, Poisson, Geometric, Negative Binomial

Why the topic needs it. Every distribution answers "what number came out, and how likely was it?". Without a symbol for "the number that came out", we cannot even ask the question. See Bernoulli trial — that is the simplest possible box.

2. The probability symbol $P (\cdot)$

The picture. Think of probability as a full bar of length 1 being sliced up among all possible outcomes. Each slice's width is that outcome's probability. All slices together fill the whole bar.

Why the topic needs it. $P (X = k) = (k n) p^{k} q^{n - k}$ and every other boxed formula is a recipe for one slice's width. The "adds to 1" rule is how we sanity-check those recipes.

3. The success/failure numbers $p$ and $q$

The picture. Cut the length-1 bar into exactly two pieces: a piece of width $p$ (the outcome labelled $1$ ) and the leftover piece of width $q$ (the outcome labelled $0$ ). That's it — that's a Bernoulli trial.

Why $q$ deserves its own letter. In formulas like $q^{k - 1} p$ we multiply " $k - 1$ failures then one success". Writing $q$ instead of $(1 - p)$ every time keeps the algebra readable — nothing more.

4. Counting: factorials $n!$ and the "choose" $(k n)$

We now need to count how many arrangements give the same number of successes. Two symbols do this.

The picture of $n!$ . You have $n$ empty slots and $n$ distinct cards. The first slot has $n$ choices, the next $n - 1$ , and so on — multiply them. Three cards → $3! = 6$ orderings.

The picture. Line up $n$ boxes. Colour exactly $k$ of them amber (successes), leave the rest white. $(k n)$ counts the distinct patterns of amber boxes.

Why the topic needs it. The Binomial's $(k n)$ counts which trials succeeded; the Negative Binomial's $(r - 1 k - 1)$ counts which of the earlier trials succeeded. Every "choose" on the parent page is exactly this box-colouring picture.

5. The sum sign $\sum$ and infinite sums

The picture. A conveyor belt drops terms $a_{1}, a_{2}, a_{3}, \dots$ into a running total. $\sum$ is the total in the bucket at the end.

Why the topic needs it. The Geometric distribution's mean $E [X] = \sum_{k = 1}^{\infty} k q^{k - 1} p$ is an infinite sum that collapses using this exact identity (differentiated once). See Geometric series for the full machinery.

6. Expectation $E [X]$ — the "average value"

The picture. Put a weight of size $P (X = k)$ at position $k$ on a number line. $E [X]$ is the balance point (centre of mass) of all those weights.

Why the topic needs it. Every distribution reports its mean: Bernoulli $p$ , Binomial $n p$ , Geometric $1/ p$ . These are the balance points. Full treatment in Expectation and Variance.

7. Variance $Var (X)$ — the "spread"

The picture. Back to the weights on the number line. Variance measures how far the weights sit from the balance point, on average (we square each distance so that weights on the left, which are "negative distances", don't cancel the ones on the right).

Why the topic needs it. Every summary row lists a variance. The parent note uses exactly this shortcut for Bernoulli (with the trick $X^{2} = X$ ). Also: variances only add when trials are independent — the reason the parent note insists on "independent trials". Details in Expectation and Variance.

8. The constant $e$ and the limit $(1 + \frac{x}{n})^{n}$

The picture. Start with £1 and add interest not once but in $n$ ever-smaller instalments per year. As $n \to \infty$ , the total approaches $e^{x}$ — continuous compounding. See Limit e^x as (1+x/n)^n.

Why the topic needs it. The Poisson formula $P (X = k) = \frac{λ ^{k} e ^{- λ}}{k !}$ is born from letting the Binomial's $(1 - \frac{λ}{n})^{n}$ turn into $e^{- λ}$ . No $e$ , no Poisson.

9. The rate $λ$

The picture. A stretch of time (or a page, or a length) with tiny random dots sprinkled on it. $λ$ is how many dots you expect in one unit stretch — this is the Poisson process viewpoint.

Why the topic needs it. $λ$ replaces the pair $(n, p)$ once $n \to \infty$ : only the product $n p = λ$ survives the limit.

Prerequisite map

The diagram below reads top-to-bottom: the random variable $X$ feeds into probability $P$ , which splits a trial into success $p$ and failure $q$ , giving a Bernoulli trial. Separately, factorials build the choose symbol. Bernoulli plus choose gives the Binomial; Bernoulli plus the sum sign and geometric series give the Geometric / Negative Binomial. The Binomial pushed to its limit produces $e$ , which (with the rate $λ$ ) yields the Poisson. Finally expectation and variance feed all five means and variances.

Equipment checklist

Cover the right side and answer aloud before revealing.

What does

x^{k}

mean for a whole number

k

Multiply

x

by itself

k

times;

x^{0} = 1

and

x^{- k} = 1/ x^{k}

What does

X

mean, and how is it different from

k

X

is the random machine (a rule turning outcomes into numbers);

k

is one specific number it outputs.

What must all the values of

P (X = k)

add up to?

Exactly

1

— the slices fill the whole probability bar.

What two numbers does a single trial's

X

take, and what do they mean?

1

= success and

0

= failure;

X \in {0, 1}

and nothing else.

p

is the success chance, what is

q

q = 1 - p

, the failure chance; pure shorthand, no new info.

For which

n

and

k

(k n)

defined, and what if

k

is out of range?

Non-negative whole numbers with

0 \leq k \leq n

; outside that range

(k n) = 0

What does

(k n)

count, in a picture?

The number of distinct ways to colour

k

n

boxes as successes, order ignored.

Write

(k n)

in terms of factorials.

\frac{n !}{k ! ( n - k )!}

What does

∣ x ∣

mean, and what does

∣ x ∣ < 1

say?

The size of

x

ignoring sign;

∣ x ∣ < 1

means

- 1 < x < 1

What is

\sum_{k = 0}^{\infty} x^{k}

for

∣ x ∣ < 1

\frac{1}{1 - x}

(the geometric series).

Over which values of

k

does

E [X] = \sum_{k} k P (X = k)

run?

Over every value

X

can take (its full support) — finite or infinite.

What is

E [c]

for a constant

c

, and

E [c X]

E [c] = c

and

E [c X] = c E [X]

— constants pass through the average.

Why does the shortcut

Var (X) = E [X^{2}] - (E [X])^{2}

hold?

Expand

(X - μ)^{2}

and average; the constant

μ

collapses the middle terms into

- μ^{2}

E [X]

is the balance point of what?

Weights of size

P (X = k)

placed at position

k

on the number line.

Why can the Binomial mean be written down instantly as

n p

Linearity of expectation — averages add even without independence, and it is

n

Bernoulli copies.

What extra condition do you need for variances to add?

Independence of the trials.

What does

n \to \infty

mean, and what is a limit?

n

grows without stopping; a limit is the fixed number a sequence settles onto.

What does

(1 + \frac{x}{n})^{n}

approach as

n \to \infty

e^{x}

What single number replaces

(n, p)

in the Poisson limit, and how?

λ = n p

, held fixed as

n \to \infty

p \to 0

Ready? Every symbol on the parent note now has a plain-words meaning and a picture behind it.