The distributions you have to know cold

A small table, known cold, converts a large share of interview probability questions into arithmetic. The skill being tested is recognition — seeing that "how many trials until the first success" is geometric — far more often than derivation.

Distribution	Describes	Mean	Variance
Bernoulli $(p)$	One trial, success or not	$p$	$p(1-p)$
Binomial $(n,p)$	Successes in $n$ independent trials	$np$	$np(1-p)$
Geometric $(p)$	Trials until the first success	$1/p$	$(1-p)/p^2$
Negative binomial $(r,p)$	Trials until the $r$ -th success	$r/p$	$r(1-p)/p^2$
Poisson $(\lambda)$	Count of rare events in a fixed window	$\lambda$	$\lambda$
Hypergeometric	Draws without replacement	$n\tfrac{K}{N}$	Below binomial
Uniform on $1..n$	A fair die	$\tfrac{n+1}{2}$	$\tfrac{n^2-1}{12}$

Table 2.1 · Discrete. The Poisson having mean equal to variance is worth remembering on its own — it is how you spot count data that is over-dispersed.

Distribution	Describes	Mean	Variance
Uniform $(a,b)$	No preference within a range	$\tfrac{a+b}{2}$	$\tfrac{(b-a)^2}{12}$
Exponential $(\lambda)$	Waiting time for a memoryless event	$1/\lambda$	$1/\lambda^2$
Normal $(\mu,\sigma^2)$	Sums of many small independent effects	$\mu$	$\sigma^2$
Lognormal	A price after multiplicative returns	$e^{\mu + \sigma^2/2}$	—
Cauchy	A ratio of two normals	Undefined	Undefined

Table 2.2 · Continuous. Cauchy is on the list precisely because it has no mean: it is the standard counterexample when someone assumes an average must exist.

Derivation 2.3

Where the die variance comes from

Worth being able to produce, because it is the one people forget.

$\mathbb{E}[X] = \frac{n+1}{2}$
$\mathbb{E}[X^2] = \frac{1}{n}\sum_{i=1}^{n} i^2 = \frac{(n+1)(2n+1)}{6}$
$\operatorname{Var}(X) = \frac{(n+1)(2n+1)}{6} - \left(\frac{n+1}{2}\right)^2$

\operatorname{Var}(X) = \frac{n^2 - 1}{12} \quad\Rightarrow\quad \text{a } d6 \text{ has } \frac{35}{12} \approx 2.92

Proposition 2.4

Memorylessness

The geometric and the exponential are the only distributions with the property that having waited does not change how much longer you expect to wait: $\Pr(X > s + t \mid X > s) = \Pr(X > t)$ . A question that says "given nothing has happened in the first ten minutes" is usually testing exactly this.

Holds when

Geometric is the discrete case, exponential the continuous one.
It is why the expected additional wait for a six, given no six yet, is still six rolls.

Equation 2.5

Normal tail figures

And in the other direction, the z-scores for the intervals you are asked to quote:

\Pr(|Z| < 1) \approx 68\%, \quad \Pr(|Z| < 2) \approx 95\%, \quad \Pr(|Z| < 3) \approx 99.7\%

$z_{50\%} = 0.67$: Half the mass lies within two thirds of a deviation.
$z_{90\%} = 1.645$: The one you need most often for a confidence interval.
$z_{95\%} = 1.96$: Two deviations, near enough, under time pressure.
$z_{99\%} = 2.58$: For the rare question that asks for a 99% interval.

The distributions you have to know cold

After this lesson you should be able to

Where the die variance comes from

Memorylessness

Normal tail figures

The rest of this lesson is in Premium