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Uniform and Normal Distributions

Uniform: flat over an interval

Every value in [a,b][a,b] is equally likely:

The uniform density
f(x)=1b−afor a≤x≤bf(x) = \frac{1}{b-a} \quad \text{for } a \leq x \leq b

Flat across its interval, and the simplest case against which to sanity-check any reading.

E[X]=a+b2Var(X)=(b−a)212E[X] = \frac{a+b}{2} \qquad \text{Var}(X) = \frac{(b-a)^2}{12}

The uniform is the case where every reading is a ratio of lengths, which makes it the one to check an intuition against before trusting it on a shape that curves.

The 112\frac{1}{12} is worth remembering, because it appears whenever someone asks for the variance of a uniform draw and it is not guessable.

The uniform is the distribution of maximum ignorance on a bounded range, and it is the raw material of simulation: every random number generator produces uniforms, and the inverse transform turns them into anything else.

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