Editing Range (statistics) (section)

===Derivation of the distribution===

Please note that the following is an informal derivation of the result. It is a bit loose with the calculation of the probabilities.

Let <math>m, M</math> denote respectively the min and max of the random variables <math>X_1 \dots X_n</math>.

The event that the range is smaller than <math>T</math> can be decomposed into smaller events according to:

* the index of the minimum value
* and the value <math>x</math> of the minimum.

For a given index <math>i</math> and minimum value <math>x</math>, the probability of the joint event:

# <math>X_i</math> is the minimum,
# and <math>X_i=x</math>,
# and the range is smaller than <math>T</math>,

is:<math display="block">
g(x) \left[ G(x+T) - G(x) \right]^{n-1}
</math>Summing over the indices and integrating over <math>x</math> yields the total probability of the event: "the range is smaller than <math>T</math>" which is exactly the cumulative density function of the range:<math display="block">
F(t) = n \int_{-\infty}^{\infty} g(x) \left[G(t+x)-G(x) \right]^{n-1} \, \text{d}x
</math>which concludes the proof.