On $\Omega=(a,b), −∞ < a < b < ∞$, each subinterval is assigned a probability proportional to the length of the interval. Find a necessary and sufficient condition for two events to be independent.
My Attempt:
For any interval $I, P(I) = \alpha .L(I)$ where $L(I)$ represents length of the interval and $\alpha$ is some cosntant.
To find the $\alpha$, $$1 = \int_a^b\int_0^y \alpha . (y-x)dxdy$$ $$1 = \alpha \int_a^b \frac{y^2}{2}dy$$ $$1 = \frac{\alpha}{6}(b^3 - a^3)$$ $$\alpha = \frac{6}{(b^3 - a^3)}$$
To prove the independence, I can start with two disjoint intervals, put them in independence equation and try to equate both sides. $$P(I_1, I_2) = P(I_1)P(I_2)$$ $$P(I_1, I_2) = \frac{6}{(b^3 - a^3)}L(I_1) . \frac{6}{(b^3 - a^3)}L(I_2)$$ $$P(I_1, I_2) = \frac{36}{(b^3 - a^3)^2}L(I_1)L(I_2)$$
Now, I'm stuck at this point I don't know how to handle LHS.