Integration to find expected value, support

Question

I am looking to find the expected value of $Y_{1}$ = E($y$) = E($y*e^{-10(y-\theta)}$) given the PDF and support $\theta \le y \lt \infty$

I.e. I need to find:

$\int_{\theta}^{\infty} 10y*e^{-10(y-\theta)} dy $

The anticipated answer is $\theta + 1/10 $ but have been struggling to get that answer using integration by parts and a change of variables.

Can anyone provide a hint?

Answer 1

here is my hint: Let $10(y-\theta) = t$ so that $ 10dy=dt$ and $y = \theta + t/10$. Now use thsese substitutions, integrating by parts and using the fact that $\lim_{t\rightarrow\infty}t/e^t = 0$ by L'hospital rule at the end, you will get your ans.