Expected value of the square of the sum of two uniformly distributed independent random variables

Question

What is $E[(X + Y)^2]$, where $X,Y ~ U(0,1)$ and independent ?

edit: I have tried expanding the brackets which results in the following: $E[X^2 + 2XY + Y^2]$, can you write this expression as $E[X^2] + 2E[XY] + E[Y^2]$ ?

Answer 1

Assuming that $X \perp\!\!\!\!\!\!\perp Y$ you expansion is

$$\mathbb{E}[X+Y]^2=\mathbb{E}[X^2]+\mathbb{E}[Y^2]+2\mathbb{E}[X]\mathbb{E}[Y]$$

Furthermore being

$$\mathbb{E}[X^2]=\mathbb{V}[X]+\mathbb{E}^2[X]$$

...it is enough to substitute to get the result