Compute Expectation for $X_{1}^\frac{1}{2}$ with Family $f(x,\theta)$

[Fermat1]

consider a density family $f(x,\theta)=\frac{exp(-{\sqrt{x}}/{\theta})}{2{\theta}^2}$.

Let $X_{1}$ have the density above. Compute $E(X_{1}^\frac{1}{2})$.

Integration by parts doesn't work since the derivative of ${\sqrt{x}}$ never vanishes, so how do I compute the expectation?

 

[stainburg]

Have you got this?

\(\displaystyle E(X_1^{1/2})=\int_0^{\infty} \sqrt{x}\exp(-\sqrt{x}/\theta)/2\theta^2dx=\int_0^{\infty} t^2\exp(-t/\theta)/\theta^2dt\)

 

[Fermat1]

Thanks, I'm a bit out of practice with integrals