Answer to Question #87369 in Statistics and Probability for Shivam Nishad

Question

Let X1, X2, ...,Xn X , X be a random sample from a distribution with density function
f(x,θ) = {θ^-1 e^-x/θ, if X>0
{0 , otherwise
Show that X=n^-1 ∑Xi is unbiased for θ .

Answer 1

The first moment of X is 

$$\mu_1=E(X)=\displaystyle\intop_{0}^\infin{x \over \theta}e^{-x / \theta}dx$$

$$\int udv=uv-\int vdu$$

$$u={x\over\theta}, du={1\over\theta}dx, v=-\theta e^{-x/ \theta}$$

$$\int {x \over \theta}e^{-x / \theta}dx=-xe^{-x/\theta}+\int e^{-x/\theta}dx=-xe^{-x/\theta}-\theta e^{-x/\theta}+C$$

$$\mu_1=E(X)=\displaystyle\intop_{0}^\infin{x \over \theta}e^{-x / \theta}dx=[-xe^{-x/\theta}-\theta e^{-x/\theta}]\begin{matrix} \infin \\ 0 \end{matrix}=\theta$$

Equating the sample first moment to the population first moment:  

$$\mu_1=\widehat{\mu}_1$$

$$\theta={1\over n}\displaystyle\sum_{i=1}^nX_i=\bar{X}$$

Therefore,

$$\widehat{\theta}={1\over n}\displaystyle\sum_{i=1}^nX_i=\bar{X}$$

${1\over n}\displaystyle\sum_{i=1}^nX_i=\bar{X}$  2 is an unbiased estimator of $\theta$.