Expected value of sum of uniformly distributed variables

Question

Let $X_i$ be a uniformly distributed random variable on the interval $[-0.5, 0.5]$

that is: $X_i$ ~ $U(-0.5, 0.5)$, for $i \in [1, 1500]$

How can I calculate the expected value of the sum $\sum_{i=1}^{1500} X_i$ ?

Answer 1

Hint:

In general: $$\mathbb E(X_1+\cdots+X_n)=\mathbb EX_1+\cdots+\mathbb EX_n$$

Provided that all expectations $\mathbb EX_i$ exist.

Also: $$\mathbb EaX=a\mathbb EX$$

Again provided that expectation $\mathbb EX$ exists.

From now on "expectation is linear" should be part of your luggage.