Uniform Distribution

The uniform distribution over $[a,b]$ has

• Notation
  • $\mathcal{U}(a,b)$
• Parameters
  • $a,b\in \mathbb{R}$
• CDF
  • $F(x)=\{\begin{array}{ll}0,& x<a,\\ \frac{x-a}{b-a},& a\le x\le b,\\ 1,& x>b.\end{array}$
• PDF
  • $f(x)=\{\begin{array}{ll}\frac{1}{b-a},& a\le x\le b,\\ 0,& \text{otherwise.}\end{array}$
• Mean
  • $\frac{a+b}{2}$
• Variance
  • $\frac{(b-a{)}^2}{12}$
• MGF
  • $\frac{e^{tb}-e^{ta}}{(b-a)t}$