Rectangular distribution

Notes

For [ilmath]X\sim\text{Rect}([a,b])[/ilmath] where [ilmath][a,b][/ilmath] denotes the closed interval [ilmath]\{x\in\mathbb{R}\ \vert\ a\le x\le b\} [/ilmath] we have the following:

• probability density function [ilmath]f:[a,b]\rightarrow\mathbb{R}_{\ge 0} [/ilmath] by [ilmath]f:x\mapsto\frac{1}{b-a} [/ilmath] - this can of course be extended to [ilmath]\mathbb{R} [/ilmath] by making it zero outside of [ilmath][a,b]\subseteq\mathbb{R} [/ilmath]
• cumulative density function [ilmath]F:[a,b]\rightarrow[0,1]\subseteq\mathbb{R} [/ilmath] by [ilmath]F:x\mapsto \frac{x-a}{b-a} [/ilmath] - this can also be extended by making it [ilmath]0[/ilmath] before [ilmath]a[/ilmath] and [ilmath]1[/ilmath] after [ilmath]b[/ilmath]

Properties are:

• [ilmath]\mathbb{E}[X]\eq\frac{1}{2}(a+b)[/ilmath] - the average of [ilmath]a[/ilmath] and [ilmath]b[/ilmath], unsurprisingly
• [ilmath]\text{Var}(X)\eq\frac{1}{12}(b-a)^2[/ilmath]
  • Giving [ilmath]\text{S.D}\eq\frac{1}{2\sqrt{3} }(b-a)[/ilmath], note that [ilmath]2\sqrt{3} \approx 3.4641[/ilmath]
• [ilmath]\text{Mdm}(X)\eq\frac{1}{4}(b-a)[/ilmath]

Note that the standard deviation (which has the same units as the mdm) is slightly larger than the mdm, the mdm is [ilmath]86.60\%[/ilmath] ([ilmath]4\ \text{s.f} [/ilmath]) of the sd