If $X\sim U(2,10)$, $Y\sim U(10,20)$, and $X$ and $Y$ are independent, find $\mathbb E[X^2Y+3]$.

Question

I know that since $X$ and $Y$ are independent that $E[X^2Y+3] = E[X^2]E[Y]+3$

$$2<X,10 \text{ and } 10<Y<20,$$

So I would set up the Integral

$$\int_2^{10} \int_{10}^{20} x^2 y+3[f(x,y)] \, dy \, dx$$

but I am lost on how to find the pdf to multiply the Expected value by. Thank you

Answer 1

Since $X$ and $Y$ are independent, we have $$\mathbb E[X^2Y+3] = \mathbb E[X^2]\mathbb E[Y] + 3. $$ We may compute this by \begin{align} &\left(\int_2^{10} \frac1{10-2} x^2 \ \mathsf dx\right)\left(\frac{10+20}2\right) + 3\\ &= \left[\frac1{24} x^3\right]_2^{10}\cdot15 + 3\\ &= 63. \end{align}