Question

Once a fire is reported to a fire insurance company, the company makes an initial estimate, X, of the amount it will pay to the claimant for the fire loss. When the claim is finally settled, the company pays an amount, Y, to the claimant. The comapny has determined that X and Y have the joint density functionf(x,y) = Given that the initial claim estiamted by the comapny is 2, determine the probability that the final settlement amount is between 1 and 3.

Answer 1

The probability that the final settlement amount is between 1 and 3 given that the initial claim is 2 = (2/9) = 0.2222

Explanation:

The complete question is presented in the attached image to this solution

The joint probability distribution is given as

f(x, y) = {2/[x²(x - 1)} × y^-[(2x-1)/(x-1)] for x>1 And y>1

Given that the initial claim estiamted by the comapny is 2, determine the probability that the final settlement amount is between 1 and 3.

That is, x = 2, and y ranges from 1 to 3

Inserting x = 2 into the expression, we obtain

f(y) = (1/2) × y⁻³ = (y⁻³/2)

The required probability would then be

P(1 < y ≤ 3) = ∫³₁ f(y) dy

= ∫³₁ (y⁻³/2) dy

= [y⁻²/-4]³₁

= [3⁻²/-4] - [1⁻²/-4]

= (-1/36) - (-1/4)

= (1/4) - (1/36)

= (8/36)

= (2/9) = 0.2222

Hope this Helps!!!