im just self learning probability and I have find pretty difficult to solve problems that involves composing a function.
For example, Let $X_{1},X_{2},...,X_{n}$ random variables iid and $f_{X}(x)= \frac{1}{x^{2}}$ when $x\geq 1$ and $f_{X}(x)=0$ in other cases and I have to find $\mathbb{E}[Y]$, when $Y=\sum_{i=1}^{n}logX_{i}$.
Whats the trick in case there is? I asked a friend and told me that it was a common trick but he didnt explained it and I havent been able to find it. Sorry if it is too elementary but im really stuck in here.
Hope you guys can help, thanks so much in advance. :D
By linearity of expectation, you have $$\mathbb{E}[Y] = \sum_{i=1}^n \mathbb{E}[\log X_i] = n\cdot \mathbb{E}[\log X_1] = n \int_{\mathbb{R}} dx f_X(x) \log x$$ the second-to-last equality since $X_1,\dots,X_n$ are identically distributed, and the last by the Law of the unconscious statistician.
Now, $$ \int_{\mathbb{R}} dx f_X(x) \log x = \int_{1}^\infty dx \frac{\log x}{x^2} = 1 $$ by evaluating the integral.