Let $X_1,...,X_n$ have the common pdf $$f(x)=\frac{1}{2\theta}\exp\left(-\frac{|x|}{\theta}\right)$$, where $x$ can be any real number and $\theta$ is positive.
I’m trying to construct the best linear unbiased estimator (BLUE) for $\theta$, and to do so, I first found $E[X^2]$, since $\theta$ is a scale parameter. After some calculus, I arrive at $E[X^2]=2\theta^2$. Now, because I want an unbiased estimator for $\theta$, I need to change the argument of $E[.]$ so that the expected value equals $\theta$.
How do I do this? The expectation operator is linear, and I don’t know how to get rid of the square.