Q4 ### Conditional and Joint Probability Distributions of Tire Pressures#### Given Problem: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 probability density function (pdf) given by:\[ f(x, y) = \begin{cases} K(x^2 + y^2) & \text{for } 22 \leq x \leq 32, \, 22 \leq y \leq 32 \\ 0 & \text{otherwise} \end{cases}\]Where \( K \) is a constant.#### (a) Determine the conditional pdf of \( Y \) given that \( X = x \):\[ f_{Y|X}(y|x) = \_\_\_\_\_\_\_\_\_\_\text{for } 22 \leq y \leq 32 \]#### Determine the conditional pdf of \( X \) given that \( Y = y \):\[ f_{X|Y}(x|y) = \_\_\_\_\_\_\_\_\_\_\text{for } 22 \leq x \leq 32 \]#### (b) Calculate the Probability:If the pressure in the right tire is found to be 22 psi, what is the probability that the left tire has a pressure of at least 25 psi? (It is known that \( K = \frac{3}{442,400} \)). Round your answer to three decimal places.\[ \text{Probability} \_\_\_\_\_\_\_\_\_\_ \]Compare this to \( P(Y \geq 25) \). (Round your answer to three decimal places.)\[ P(Y \geq 25) = \_\_\_\_\_\_\_\_\_\_ \]#### (c) Expectation and Standard Deviation:If the pressure in the right tire is found to be 22 psi, what is the expected pressure in the left tire, and what is the standard deviation of pressure in this tire? (It is known that \( K = \frac{3}{442,400} \)).Round your answers to two decimal places.

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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) 22 ≤ x ≤ 32, 22 ≤ y ≤ 32 otherwise (a) Determine the conditional pdf of Y given that X = x. frix(xlx) = for 22 ≤ y ≤ 32 Determine the conditional pdf of X given that Y = y. fx|x(xly) = for 22 ≤ x ≤ 32 3 (b) If the pressure in the right tire is found to be 22 psi, what is the probability that the left tire has a pressure of at least 25 psi? (It is known that places.) = Round your answer to three decimal 442,400 Compare this to P(Y ≥ 25). (Round your answer to three decimal places.) P(Y ≥ 25) = 3 (c) If the pressure in the right tire is found to be 22 psi, what is the expected pressure in the left tire, and what is the standard deviation of pressure in this tire? (It is known that K = Round your answers to two decimal places.) 442,400 expected pressure psi standard deviation psi