Rectangular distribution
From Maths
Notes
For X∼Rect([a,b]) where [a,b] denotes the closed interval {x∈R | a≤x≤b} we have the following:
- probability density function f:[a,b]→R≥0 by f:x↦1b−a - this can of course be extended to R by making it zero outside of [a,b]⊆R
- cumulative density function F:[a,b]→[0,1]⊆R by F:x↦x−ab−a - this can also be extended by making it 0 before a and 1 after b
Properties are:
- E[X]=12(a+b) - the average of a and b, unsurprisingly
- Var(X)=112(b−a)2
- Giving S.D=12√3(b−a), note that 2√3≈3.4641
- Mdm(X)=14(b−a)
Note that the standard deviation (which has the same units as the mdm) is slightly larger than the mdm, the mdm is 86.60% (4 s.f) of the sd