Common distributions (Bernoulli, Binomial, Poisson): Foundations

This page assumes you have seen nothing. We build each symbol from a picture, in the order the topic needs them. If you already know some, skim to the parent topic and come back when a symbol bites you.

0. What is a "random variable"? (the symbol $X$ )

Every formula on the parent page starts with $X$ . So what is it?

Picture a coin flip. The physical outcome is "heads" or "tails" — those are words, not numbers. We choose to record heads as the number $1$ and tails as the number $0$ . That translation is exactly what $X$ does.

Figure — Common distributions (Bernoulli, Binomial, Poisson)

Why do we need $X$ ? Because you cannot do arithmetic — averages, spreads, sums — on the words "heads" and "tails". You can do arithmetic on $0$ and $1$ . The whole topic is arithmetic on outcomes, so every outcome must first become a number.

1. Probability $P (\cdot)$ — a number between 0 and 1

Picture a bar of length $1$ . You chop it into pieces, one piece per possible outcome. The length of a piece is that outcome's probability. Because the whole bar has length $1$ , all the pieces must add to $1$ .

Why does the topic need this? Every distribution is nothing more than a rule for how to chop that bar — how much length to give each possible count of successes.

2. The symbol $p$ (and $1 - p$ )

So $p$ and $1 - p$ are just the two pieces of the length- $1$ bar for a single yes/no experiment. If success takes up $0.3$ of the bar, failure must take the remaining $0.7$ .

3. Summation $\sum$ — "add up a list"

The mean and variance derivations are stuffed with $\sum$ . Here is all it means.

Picture a checklist. You walk down the list of possible values, compute one number per row, and total the column at the bottom.

$\sum_{x \in {0, 1}} x = 0 + 1 = 1, \sum_{k = 1}^{3} k^{2} = 1 + 4 + 9 = 14$

Why the topic needs it: the expected value is a weighted sum over every outcome, and a sum is exactly what $\sum$ writes compactly.

4. Expected value $E [X]$ — the balance point

Picture the chopped bar from §1, but now stand each piece up as a weight on a seesaw, placed at its value $x$ . $E [X]$ is the point where the seesaw balances.

Why "value times probability"? A value that almost never happens should barely tug the balance point; a value that happens often should tug hard. Multiplying by probability is exactly that weighting.

5. Variance $Var (X)$ and the square

Picture two seesaws that both balance at the same point. One has its weights bunched near the middle (small variance); the other has them flung to the ends (large variance). Same balance point, different spread.

Why square? Distances above and below the mean would cancel if we just added them. Squaring makes every deviation positive, so they accumulate instead of cancelling. $E [X^{2}]$ means "average of the squared values" — apply the seesaw rule to $x^{2}$ instead of $x$ .

Recall Why variance peaks at

$p = 0.5$ $Var = p (1 - p)$ is a downward parabola in $p$ with maximum where? ::: At $p = 0.5$ , where $p (1 - p) = 0.25$ — maximum uncertainty, both outcomes equally likely.

6. The factorial $n!$ and the binomial coefficient $(k n)$

The Binomial PMF hinges on $(k n)$ . Build it from the factorial.

Why is $0! = 1$ ? There is exactly one way to arrange nothing — the empty arrangement — so the count is $1$ , not $0$ . This matters: the $k = 0$ and $k = n$ ends of the Binomial rely on it.

Picture $n$ empty boxes and you must colour exactly $k$ of them. Each colouring pattern is one "arrangement" of successes. $(k n)$ is how many distinct patterns exist. That is precisely the parent's "which 8 of the 100 emails got clicked" count.

7. Powers $p^{k}$ and the exponential $e^{- λ}$

Why multiply and not add? Picture branching paths: after one success ( $p$ of them), the survivors split again ( $p$ of those), and so on. Fractions of fractions shrink by multiplication.

Why does the topic need $e$ ? When you repeat "tiny survival chance" a huge number of times, the product does not blow up or vanish arbitrarily — it converges to a clean value built from $e$ . That convergence is exactly what turns Binomial into Poisson.

8. The arrow $\to$ and "limit" (how Poisson is born)

Picture a sequence of numbers creeping closer and closer to a target line but the individual steps never quite land — the target is the limit. The parent's Poisson derivation watches three such creeping quantities settle to $1$ , $e^{- λ}$ , and $1$ .

Prerequisite map

Equipment checklist

Test yourself — say each answer aloud before revealing.

What does big

X

mean versus little

x

X

is the outcome-to-number machine;

x

is one specific number it produced.

What two rules must every probability obey?

Each is between

0

and

1

; all of them add to exactly

1

If success has probability

p

, what is failure's probability?

1 - p

, because the two pieces fill the whole length-

1

bar.

In words, what does

\sum_{x}

tell you to do?

Plug in every allowed value of

x

, compute the expression, and add all results.

How do you compute

E [X]

from a table of values and probabilities?

Multiply each value by its probability and add them all up.

Why do we square deviations when finding variance?

So above and below the mean do not cancel; squaring makes every deviation positive.

What is

0!

and why?

0! = 1

, because there is exactly one way to arrange nothing.

In plain words, what does

(k n)

count?

The number of ways to choose

k

items out of

n

when order does not matter.

Why do independent probabilities multiply (giving

p^{k}

Each new success is a fraction of the survivors, and fractions of fractions combine by multiplication.

What does

n \to \infty

actually mean?

The value the expression settles toward as

n

grows without bound — not literally infinity.

What role does

λ

play in Poisson?

The average count of events; it is the Poisson analogue of

n p

Ready? Head back to the parent topic and every symbol will now be a friend, not a wall.