An insurance agent receives a bonus if the loss ratio L on his business is less than 0.5, where L is the total losses (say, X) divided by the total premiums (say, T). The bonus equals (0.5 − L)(T/30) if L < 0.5 and equals zero otherwise. If X (in $100,000) has the pdf

and if T (in $100,000) equals 3, determine the expected value of the bonus.

Answer :

Step 1 of 1:

Given the loss ratio L on his business is less than 0.5.

Where L is the total losses divided by the total premiums.

Let B(x) = the bonus received given that the total losses was x.

Then since X=LT and T=3.

We have x = total losses, whose pdf is

Our goal is to find

The expected value of the bonus.

We have x = total losses, whose pdf is

We know that

B(x) = the bonus received given that the total losses was x.

Then since X=LT and T=3.

The formula for B(x) is given by,

Then...