Uniform Distribution

Uniform Distribution §

def. Uniform Distribution. $X$ has uniform distribution of it has a uniform density on interval $(a,b)$:

$$\begin{array}{r}X\sim \text{Unif(a,b)}\\ f_X(x)=\{\begin{array}{ll}\frac{1}{b-a}& x\in (a,b)\\ 0& \text{else}\end{array}\\ F_X(t)={\int }_{-\infty }^t{f}_X(x)dx=\{\begin{array}{ll}\frac{t-a}{b-a}& t\in (a,b)\\ 1& t>b\\ 0& t<a\end{array}\end{array}$$ $$\mathbb{E}(X)=\frac{a+b}{2}\mathbb{SD}(X)=\frac{b-a}{2\sqrt{3}}$$

Estimators §

let

• $X\sim \text{Unif}(a,b)$
• $X_1,\dots ,X_n\stackrel{iid}{\sim }\text{Unif}(a,b)$

Log likelihood:

$$\ln {\mathcal{L}}_n(a,b|x_1,...,x_n)=\ln \frac{1}{(b-a{)}^n}$$

score:

$$s_n(a)=\frac{n}{b-a}$$ $$s_n(b)=\frac{-n}{b-a}$$

MLEs:

$${\hat{a}}_{\text{MLE}}=\text{min}(X_1,...,X_n)$$ $${\hat{b}}_{\text{MLE}}=\text{max}(X_1,...,X_n)$$