Mean of a uniform distribution where the limits are themselves distributed.

Question

I am trying to work out $E[\pi] $ where $\pi\sim U[k_1 , k_2] $. Where $k_1 = 0.5 - \omega_1$ and $k_2 = \omega_2 + 0.5$ and $\omega_1$ and $\omega_2$ are independent random draws from a uniform distribution $[0,0.5) $. The value of $E [\pi] = 0.5$. But I don't know how to get there. I know that the mean of a uniform distribution is $\frac{a-b}{2}$. But I don't know how to use this when the limits are themselves distributed.

Answer 1

You know that

$$E[\pi\vert w_1,w_2] = \frac{k_1+k_2}{2} = \frac{1-w_1+w_2}{2}$$

Lets take the expectation of this quantity regarding the pair $(w_1,w_2)$.

$$E_{(w_1,w_2)}E[\pi\vert w_1,w_2] = E_{(w_1,w_2)}\frac{1-w_1+w_2}{2} = \frac12$$ as $E_{(w_1,w_2)}(w_1) = E_{(w_1,w_2)}(w_2)$.