I don't get how to calculate a conditional expected value. Here's the setting: $v_s$ and $v_b$ are independent and both uniformly distributed on [0,1]. Furthermore, $p_s(v_s)=a_s +c_sv_s$ and $p_b(v_b)=a_b+c_bv_b$.
Now I am asked to calculate $E[p_s(v_s)|p_s(v_s)≤p_b(v_b)]$ and $E[p_b(v_b)|p_b(v_b)≥p_s(v_s)]$.
Can anyone please help me?
I would really appreciate any help!
If you have a uniform random variable $V$ on the interval $[a,b]$ and you want to compute the conditional expected value of $V$ given $V<C,$ where $C$ is a random variable independent of $V$, there are three cases of interest.
First, if $C>b,$ then the inequality $V<C$ gives you no information that you don't already know since you already know $V<b.$ So in that case the conditional expectation is just the regular expectation $(b+a)/2.$
If $C<a,$ then it is impossible that $V<C,$ so the conditional expectation is undefined.
If $a\le C\le b$ then you know that $a < V < C$ so $V$ is conditionally uniform in $[a,C]$ which means that the conditional expected value is $ (a+C)/2.$
So in your original problem, if the constants are picked so that $a_s < p_b < a_s+c_s$ always hold regardless of what $v_b$ is, then the third case gives $E(p_s | p_s < p_b) = (a_s+p_b)/2.$