91Ó°ÊÓ

A random variable is a fundamental concept in statistics and probability theory. It represents a variable whose possible values are numerical outcomes of a random phenomenon. In this exercise, we have two random variables:

• \( X \) for the air pressure in the right tire.
• \( Y \) for the air pressure in the left tire.

The joint probability density function (pdf) \( f(x, y) = K(x^2 + y^2) \) describes the probability of \( X \) and \( Y \) occurring together within a specified range (from 20 to 30 psi). A joint pdf allows us to assess the likelihood of certain combinations of values for two or more random variables.

The task in the exercise, in part, is to determine the probability that the tire pressures meet certain conditions, such as being underfilled. When working with random variables and their pdfs, it is crucial to understand not only individual outcomes but also how they combine and interact.

In probability, two random variables are considered independent if the occurrence of one does not affect the probability of the occurrence of the other. Mathematically, \( X \) and \( Y \) are independent if their joint pdf can be factored into the product of their marginal pdfs:
\[f(x, y) = f_X(x) \times f_Y(y)\]
To verify independence, the exercise requires the calculation of the marginal distributions, \( f_X(x) \) and \( f_Y(y) \), by integrating out the other variable. If these products equal the joint pdf, \( X \) and \( Y \) are independent.
In this particular exercise, the joint pdf cannot be straightforwardly decomposed into such a product, indicating that \( X \) and \( Y \) are likely not independent. This outcome suggests that changes in pressure in one tire may indeed affect the probability distribution of pressure in the other tire.

The marginal distribution provides the probabilities of a single random variable irrespective of the values of other random variables. It is obtained by integrating the joint pdf over the other variable.
For the right tire's random variable \( X \), its marginal distribution \( f_X(x) \) is derived as follows:\[f_X(x) = \int_{20}^{30} K(x^2 + y^2) \, dy\]
This integral sums over the range of all possible values of \( Y \) to find the distribution of \( X \) alone.
Similarly, to find the marginal distribution of \( Y \), integrate over \( X \):\[f_Y(y) = \int_{20}^{30} K(x^2 + y^2) \, dx\]
These integrals need to be solved to provide the marginal distribution functions, revealing important information on the distribution of air pressure individually in each tire without considering the other's influence.

Underfilled probability refers to the likelihood that a tire's air pressure falls below the recommended level, in this case, 26 psi. In our exercise, we are interested in the probability of both tires being underfilled together which involves evaluating a specific region under the joint pdf.
To find this probability:\[P(X < 26, Y < 26) = \int_{20}^{26} \int_{20}^{26} K(x^2 + y^2) \, dy \, dx\]
By performing this double integration with the value of \( K \) determined earlier, we can calculate the likelihood that both tires are underfilled.
This calculation gives us a practical insight into the risk of driving with tires that are not inflated to a safe level. Understanding such probabilities helps in evaluating how often corrective measures like refilling might be necessary to ensure safety and optimal performance.}]}