how to know if an estimator is unbiased

Question

I have some doubt about this excercise, can someone help me?

Let $(X_1,\dots,X_n)$ be independent identically distributed random variables with p.d.f. $$f(x) = \theta^2 x \exp(-\theta x),$$ with $x>0$.

Is $T(X_1,\dots,X_n)= 1/X_1$ an unbiased estimator of $\theta$?

I know that this estimator is unbiased when $E(X)$ is equal to $\theta$, but how can I find $E(X)$ of an estimator it's different from $E(X)$ of a random varible ?

Answer 1

Hint :

Using the law of the unconscious statistician you get that

$$E[\frac{1}{X}] = \int_{0}^{\infty} \frac{f(x)}{x} dx$$