information:extraction, linguistic:paraphrasing
Probability density function

The probability density function of the continuous uniform distribution is

The values of f (x) {\displaystyle f(x)} at the two boundaries a {\displaystyle a} and b {\displaystyle b} are usually unimportant, because they do not alter the value of ∫ c d f (x) d x {\textstyle \int _{c}^{d}f(x)dx} over any interval, {\displaystyle,} nor of ∫ a b x f (x) d x, {\textstyle \int _{a}^{b}xf(x)dx,} nor of any higher moment. Sometimes they are chosen to be zero, and sometimes chosen to be 1 b − a. {\displaystyle {\tfrac {1}{b-a}}.} The latter is appropriate in the context of estimation by the method of maximum likelihood. In the context of Fourier analysis, one may take the value of f (a) {\displaystyle f(a)} or f (b) {\displaystyle f(b)} to be 1 2 (b − a), {\displaystyle {\tfrac {1}{2(b-a)}},} because then the inverse transform of many integral transforms of this uniform function will yield back the function itself, rather than a function which is equal " almost everywhere ", i.e. except on a set of points with zero measure. Also, it is consistent with the sign function, which has no such ambiguity.

Any probability density function integrates to 1, {\displaystyle 1,} so the probability density function of the continuous uniform distribution is graphically portrayed as a rectangle where b − a {\displaystyle b-a} is the base length and 1 b − a {\displaystyle {\tfrac {1}{b-a}}} is the height. As the base length increases, the height (the density at any particular value within the distribution boundaries) decreases.

In terms of mean μ {\displaystyle \mu } and variance σ 2, {\displaystyle \sigma ^{2},} the probability density function of the continuous uniform distribution is