Suppose that $X_1, X_2, X_3$ denote a random sample of size three from a continuous distribution with probability density function (pdf)
$f_X(x) = \frac{1}{\theta}e^{\frac{−x}{\theta}}, x>\theta.$
Consider the following three estimators for $\theta:\hat\theta_1 = X_1; \hat\theta_2=\frac{X_1+X_2+X_3}{3} = \bar X;\theta_3 =\frac{X_1+2X_2}{3}$. Given that the moment generating function (mgf) of $X$ is $M_X(t) = \frac{1}{1−\theta t}.$
How do I find Find $E(X)$ and $E(X^2)$ and determine which estimators for $\theta$ are unbiased.
To derive $\mathsf E[X^n]$ from $M_X(t)$ (aka the $n^{th}$ moment of $X$), you take $\cfrac{\mathrm d^n}{\mathrm dt^n} M_X(t)$ and evaluate the result at $t=0.$
$$\cfrac{\mathrm d}{\mathrm dt} \cfrac{1}{1-\theta t} = \cfrac{\theta}{(1-\theta t)^2}$$
Evaluating at $t=0$ gives:
$$\mathsf E[X] = \theta$$
I'll let you find $\mathsf E[X^2].$ To determine which of the given estimators are unbiased, you should know that an estimator $\hat\theta$ for $\theta$ is unbiased when $\mathsf E[\hat \theta] = \theta$. For example,
$$\mathsf E[\hat\theta_1] = \mathsf E[X_1]= \mathsf E[X] = \theta$$
so $\hat\theta_1$ is an unbiased estimator for $\theta$. Hopefully this helps you with the other two.