Final Answer:
(a) The covariance of the number of failures for the machines that make widgets is Cov(X, Y) = -15.
(b) No, the random variables are not independent.
Step-by-step explanation:
In order to find the covariance of the random variables X and Y, we use the formula: Cov(X, Y) = E[XY] - E[X]E[Y]. Given the joint density function of X and Y, we can calculate the expected values required for this formula. The joint density function provides the probability distribution of the two variables together. By integrating over the joint density function for the product XY, we find E[XY]. Similarly, we find E[X] and E[Y] by integrating over the marginal density functions. Substituting these values into the covariance formula, we get Cov(X, Y) = -15.
The negative covariance indicates an inverse relationship between the number of failures for the machines that make widgets. When one variable increases, the other tends to decrease. Now, to determine if the random variables are independent, we can use the fact that independent random variables have a covariance of zero. Since Cov(X, Y) is non-zero (-15 in this case), we conclude that X and Y are not independent.
In practical terms, this implies that the occurrence of failures in one machine affects the likelihood of failures in another, suggesting some form of interdependence or shared factors affecting both machines. Understanding the covariance and independence of these variables is crucial for predicting and managing the reliability and performance of the machines in widget production.