A company offers a basic life insurance policy to its employees, as well as a supplemental life insurance policy. To purchase the supplementa policy, an employee must first purchase the basic policy. Let X denote the proportion of employees who purchase the basic policy, and Y the proportion of employees who purchase the supplemental policy. Let X and Y have the joint density function f(x,y)= 2(x+y). Given that 10% of the employees buy the basic policy, what is the probability that fewer than 5% buy the supplemental policy?

This looks like an actuary problem.

Anyway, the jpf is 2(x+y),

0<y<x<1 (why?).

2\int_{0}^{x}(x+y)dy=3x^{2}, \;\ 0<x<1

f(y|x)=\frac{2(x+y)}{3x^{2}}, \;\ 0<y<x

f(y|x=\frac{1}{10})=\frac{2(\frac{1}{10}+y)}{3( \frac{1}{10})^{2}}=\frac{20(10y+1)}{3}, \;\ 0<y<\frac{1}{10}

P(Y<\frac{1}{20}|X=\frac{1}{10})=\frac{20}{3}\int_{0}^ {\frac{1}{20}}(10y+1)dy

Finish the integration and you have it.

Thank u very much