Question

Each rear tire on an experimental airplane is supposed to be filled to a pressure of 42 pounds per square inch (psi). Let X denote the actual air pressure for the right tire and Y denote the actual air pressure for the left tire. Suppose that X and Y are random variables with the joint density function shown below. Complete parts (a) through (c). f(x,y)={ k(x 2 +y 2 ), 0, ​ 28≤x<56,28≤y<56 elsewhere ​ (a) Find k. k= (b) Find P(45≤X≤53 and 34≤Y≤37). P(45≤X≤53 and 34≤Y≤37)= (Simplify your answer.) (c) Find the probability that both tires are underfilled. The probability that both tires are underfilled is (Simplify your answer.)

Answer 1

To answer the student's questions, one has to calculate the normalization constant k by integrating the given joint density function over the range of pressures where the function is defined, and setting the total probability to 1. For b and c, one must perform specific integrations over the given ranges to find the respective probabilities.

Step-by-step explanation:

The student has asked about an experimental airplane with rear tires filled to a certain air pressure. In this scenario, the actual air pressures for the right and left tires are denoted by random variables X and Y, respectively. They have a joint density function provided with certain conditions for the values of X and Y within the range of 28 to 56 psi, and the values outside this range are considered 0. Let's break down the questions:

Part (a): Find k

To find the value of k, we need to ensure that the total probability over the valid range of X and Y equals 1, which involves integrating the joint density function over the specified ranges of X and Y and setting that integral equal to 1:

`\(\int_(28)^(56)\int_(28)^(56) k(x^2 + y^2) dx~dy = 1\)`

Solving this would give us the value of k.

Part (b): Find
`\(P(45\leq X \leq 53 \text{ and } 34\leq Y \leq 37)\)`

This probability is found by integrating the joint density function over the specified ranges of X and Y:

`\(\int_(45)^(53)\int_(34)^(37) k(x^2 + y^2) dx~dy\)`

Part (c): Probability that both tires are underfilled

The tires are considered underfilled when the pressure is below 42 psi. Therefore, we want to find:

`\(P(X < 42 \text{ and } Y < 42)\)`

This probability can be determined by integrating the joint density function over the ranges from 28 to 42 for both X and Y:

`\(\int_(28)^(42)\int_(28)^(42) k(x^2 + y^2) dx~dy\)`