Can anyone figure out this actuary claim amount question?

Quote:

Originally Posted by Statsnoob2718

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 company has determined that X and Y have the joint density function
$f(x,y)=\frac{2}{x^2(x-1)}y^{-(\frac{2x-1}{x-1})}$ for x > 1, y > 1

Given that the initial claim estimated by the company is 3, determine the probability that the final settlement amount is between 2 and 4 .
$\int_1^{\infty}\int_1^{\infty}\frac{2}{x^2(x-1)}\int_1^{\infty}y^{-(\frac{2x-1}{x-1})}dy$

thank you for your help. thank you a lot

$f(y | x) = \frac{f(x, y)}{f_x(x)}$ where $f_x(x)$ is the marginal density $\int_1^{+\infty} f(x, y) \, dy$ .

Now calculate $\int_2^4 f(y | x = 3) \, dy$ .