Uniform Distribution Example_MoM

Example) Mathematical Statistics and Data Analysis, 3ED, Chapter 8. Q53(a)

Let $X_{1},X_{2},\cdots, X_{n}$ be iid uniform [0, $\theta$] Find the method of moments estimate of $\theta$ and its mean and variance. 

$\triangleright$ Solution
$f(X|\theta)=\frac{1}{\theta}$ for any x between 0 and $\theta$

$E(X)=\int_{0}^{\theta}\frac{x}{\theta}dx=\frac{\theta}{2}$ 
$E(X^2)=\int_{0}^{\theta}\frac{x^2}{\theta}dx=\frac{\theta^2}{3}$ 
$\rightarrow Var(X)=E(X^2)-[E(X)]^2=\frac{\theta^2}{3}-\frac{\theta^2}{4}=\frac{\theta^2}{12}$

$E(X)=\bar{X}=\frac{\theta}{2}\rightarrow \therefore \theta=2\bar{X} \rightarrow \hat{\theta}=2\bar{X}$ 

$\therefore E(\hat{\theta})=E(2\bar{X})=E(2\cdot \frac{1}{n} \sum x_{i})=2 \cdot \frac{1}{n} \cdot n \cdot \frac{\theta}{2}= \theta$ 

$\therefore Var(\hat{\theta})=Var(2\bar{X})=4Var(\frac{1}{n}\sum x_{i})= \frac {4}{n^2} \sum Var(X_{i}) = \frac {4}{n^2} \sum \frac{\theta^2}{12} = \frac{4}{n^2} \cdot \frac {n\theta^2}{12}=\frac{\theta^2}{3n}$ 

Uniform Distribution

Continuous Random Variable 
X~U(a,b): All intervals of the same length within a given range (a,b) is equally probable!


$\star F(x)= \frac{x-a}{b-a}$ where $a < x < b$  
$\star f(x)=\frac{1}{b-a}\cdot I_{(a,b)}(X)$ 

$\star E(X)= \frac{a+b}{2}$, $Var(X)=\frac{(b-a)^2}{12}$
Proof
$E(X)=\int_{a}^{b}x \cdot \frac{1}{b-a} dx= \frac{x^2}{2(b-a)}|_{a}^b = \frac{b^2-a^2}{2(b-a)}=\frac{a+b}{2}$
$E(X^2)=\int_{a}^{b}x^2 \cdot \frac{1}{b-a} dx= \frac{x^3}{3(b-a)}|_{a}^b = \frac{b^3-a^3}{3(b-a)}=\frac{b^2+ab+a^2}{3}$
$Var(X)=\frac{a^2+ab+b^2}{3}-(\frac{a+b}{2})^2=\frac{(b-a)^2}{12}$



Method of Moments in Uniform Distribution 
$\star$ Moment Generating Function m(t) is...
$m(t)=\int_{0}^{\theta}e^{xt}\cdot p(X=x)dt=\int_{0}^{\theta} e^{xt}\cdot \frac{1}{\theta}dt = \frac{1}{\theta}\cdot \frac{1}{x}e^{xt} |^{\theta}_{0}= \frac{1}{\theta}\cdot \frac{1}{x}\cdot e^{\theta x}$ 

Example) Mathematical Statistics and Data Analysis, 3ED, Chapter 8. Q53(a)
Find the method of moments estimate of $\theta$ and its mean and variance. 
Solution??!!