How to compute the expected value of the sum of two random variables?

Question

How do you compute expected value? I need help for not word problems, but a question similar to this:

Two numbers, $x$ and $y$, are randomly chosen from the interval $[0,1]$. What is the expected value of $(x+y)$?

Note that this is NOT the exact question. Mine has a different interval.

Answer 1

There is already a good answer here. To offer a bit more info -

In general, we can find the expected value of g(x,y) - that is, $E[G(X,Y)]$ - by using a double integral:

$$\int^\infty_{-\infty} \int^\infty_{-\infty}g(x,y)F(x,y)dxdy$$

so for E(X+Y) over [0,1] we can find:

E(y) + E(x)

$\int^1_0ydy + \int^1_0xdx$

$\frac{1}{2}y^2\Big|_0^1$ + ...

= $.5+.5$

= 1

Answer 2

By the linearity of expectation, we have

$$E(x+y)=E(x)+E(y) = 0.5 + 0.5 = 1$$