= Uniform Distribution =

The '''uniform distribution''' is a flat continuous probability distribution function. All values are equally likely.

<<TableOfContents>>

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== Description ==

The distribution is given by {{attachment:uniform.svg}} for any ''a ≤ x ≤ b''. The function is defined for all real numbers so values of ''x'' outside that range map to 0.

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== Moments ==

The [[Analysis/ExpectedValue|expected value]] is given as ''E[X] = (a+b)/2''. The derivation is:

{{attachment:exp1.svg}}

{{attachment:exp2.svg}}

[[Analysis/Variance|Variance]] is given as ''Var[X] = (b-a)^2^/12''. It is sometimes given as ''(a-b)^2^/12'' instead, but it doesn't matter since the difference is squared and any sign is discarded.

The simplest derivation is through ''Var[X] = E[X^2^] - E[X]^2^''. The second raw moment is calculated as:

{{attachment:var1.svg}}

{{attachment:var2.svg}}

Meanwhile the square of the expected value is simple to calculate:

{{attachment:var3.svg}}

Altogether now:

{{attachment:var4.svg}}



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CategoryRicottone