Final answer:
The joint pdf problem requires finding a normalizing constant K, calculating various probabilities involving the pressure in both tires, determining independence, and calculating conditional distributions. Marginal and conditional distributions are also needed to find expected values and standard deviations.
Step-by-step explanation:
Joint Density Function and Properties
To answer the student's question, we first need to find the value of the normalization constant K for the joint pdf. Since the pdf must integrate to 1 over the region of interest, we can set up the integral for K:
\[1 = \int_{20}^{30} \int_{20}^{30} K(x^2 + y^2) \, dx \, dy\]
We solve this integral to find K.
For the probability that both tires are underfilled, we define 'underfilled' and integrate the joint pdf over that region.
The probability that the difference in air pressure is at most 2 psi involves setting up and evaluating an appropriate integral with the given constraint.
The marginal distribution of X is found by integrating the joint pdf over all values of Y.
Independence of X and Y can be determined by seeing if the product of their marginal distributions equals the joint distribution.
Conditional probability density functions are found by dividing the joint pdf by the marginal pdf of the variable we are conditioning on.
Finally, the questions related to the pressure being 22 psi in the right tire involve using the conditional distribution of Y given X = 22.
The mean and standard deviation of the left tire pressure, given the pressure in the right tire, requires calculating expected value and variance using the conditional pdf.