integration expected value

Question

Let f(x) = 1/2e^(-x/2), for x greater than 0. What is the expected value of X?

Answer 1

The expected value is the ratio of A / B

where A = ∫ x exp(-x/2) dx limits 0 to ∞

and B = ∫ exp(-x/2) dx limits 0 to ∞

The antiderivative for B is easy: -2 exp(-x/2)

Using integration by parts, the antiderivative for A is: - 2 x exp(-x/2) - 4 exp(-x/2)

This leads to A/ B = 4/2 = 2