I have some homework that says:
$X$ is a real-valued stochastic variable, uniformly distributed on $(0,1)$ and let $Y=-2 \ln X$. Find $E(Y)$.
My idea is just to do:
$$E(Y)=\int_0^1 -2\cdot \ln X \, d\lambda=2$$ where $\lambda$ is the Lebesgue measure.
It this correct and it is really that simple?
Thanks.
First note that since $X \sim Uniform(0,1)$, $\mathbb{P}(X \in(0,1)) = 1$ and thus without loss of generality we can assume that $X> 0$ everywhere, so that $Y$ is well-defined.
Now, $$\mathbb{E}Y = -2 \mathbb{E} [\ln X] = -2\int_\Omega \ln X d \mathbb{P} = -2 \int_0^1 \ln x \mathbb{P}_X(dx) = -2 \int_0^1 \ln x \lambda(dx) = 2$$