Question

Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable. X for the right tire and Y for the left tire, with joint pdf.

f(x,y) = k(x² + y²), when 20 ≤ x ≤ 30, 20 ≤ y ≤ 30, and

f(x,y) = 0 (otherwise)

A. What is the value of k.

B. what is the probability that both tires are underfilled?

C. What is the marginal distribution of air pressure in the right tire?

Answer 1

A)

Let denotes the air pressure for the right tire and denotes air pressure for the left tire.

The joint probability densities function of is as follows:

The given density is a valid joint probability density function. That is,

Consider,

Now, substitute the value of in the given joint density function as follows:

The joint density function is,

The obtained value of is

B)

The probability that both tires are under filled is 0.3024.

C)

The marginal distribution of air pressure in the right tire is,