Question

Two towns, Besiktas and Sariyer are going to raise a fund together to build a new public library. Suppose Besiktas will raise B million Turkish Liras and Sariyer will raise S million Turkish Liras. B has exponential distribution with rate 1.3 and S has exponential distribution with rate 2.2. Suppose B and S are independent. The new library will be built if they can raise 4 million Turkish Liras or more. What’s the probability that Besiktas will raise less than 2 million Turkish Liras given that the library can be built?  (P(B<2 | B+S≥4)=?)

Answer 1

Since B has exponential distribution with rate 1.3 and S has exponential distribution with rate 2.2. Suppose B and S are independent.

So the joint pdf of B and S is

Now,

So,

Now by definition of conditional probability

Now,

Hence the required probablity is