I have a Manager who has no information about the profit from investment into two bonds, other than the fact that they are independently drawn from uniform distribution ${m, 1}$ where, m>0. Profit from 1st and 2nd bond are denoted by a and b respectively.
He asks his partner (who is knowledgeable about profit from two bonds but is risk averse) to invest among two bonds. Manager knows that 1st bond is less risky and so partner will invest in a when a>=t*b where 0<t<1. So, basically only when profit from a is very very low, will partner not invest in 1st bond.
Now manager sees that partner has invested in 1st bond so it must be the case that a>=t*b. So, given this information he is trying to compute expected profit.
Since t<1, and both a and b has domain space over m to 1, so it might be the case that t*b falls below m. How to handle that in integration -
I know expected value will look something like this -
$$ \frac{ \int_{m}^{1} \int_{a= Max[t*b, m]}^1 a ~da db} {\int_m^1 \int_{a= Max[t*b, m]}^1 1 ~ da db} $$
but I am not sure how can I solve for Max[t*b, m] mechanically?
Is there a simpler way to address this? Any help will be appreciated