Shape
Distribution
When to use it
Mean & Variance
Tool
Binomial A-level
X ~ B(n, p)
P(X=x) = nCx px(1-p)n-x
A fixed number n of independent trials, each with the same success probability p. You count how many succeed - e.g. 20 free throws, how many go in.
E(X)np
Varnp(1-p)
Uniform FM
X ~ U(n) over 1, 2, ..., n
P(X=x) = 1/n
Finitely many outcomes that are all equally likely - e.g. a fair die, or drawing one numbered ticket from a bag.
E(X)(n+1)/2
Var(n2-1)/12
Geometric FM
X ~ Geo(p)
P(X=x) = (1-p)x-1 p
The number of trials up to and including the first success - e.g. how many rolls of a die until the first six.
E(X)1/p
Var(1-p)/p2
Poisson FM
X ~ Po(λ)
P(X=x) = e-λλx / x!
Counting events that occur at a constant average rate in a fixed interval of time or space, independently - e.g. emails per hour, flaws per metre.
E(X)λ
Varλ
Shape
Distribution
When to use it
Mean & Variance
Tool
Continuous Uniform FM
X ~ U(a, b)
f(x) = 1/(b-a), a ≤ x ≤ b
Every value in an interval is equally likely - e.g. a bus arriving at a random moment in a 10 minute window. Also called the rectangular distribution.
E(X)(a+b)/2
Var(b-a)2/12
Exponential FM
X ~ Exp(λ)
f(x) = λe-λx, x ≥ 0
The waiting time until the next Poisson event - e.g. time until the next phone call. Memoryless: time already waited does not change the future wait.
E(X)1/λ
Var1/λ2
Normal A-level
X ~ N(μ, σ2)
f(x) = (1/σ√(2π)) e-(x-μ)²/2σ²
Symmetric data clustering around a mean, most values near the centre - e.g. heights, measurement error. Also the limit of many averages (Central Limit Theorem).
E(X)μ
Varσ2