Rectangular distribution

Notes

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

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

Properties are:

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

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