Question

let x and y be the number of hardware failures in two computer labs in a given month. the joint distribution of x and y is given in the table below. p(x, y) x 0 1 2 0 0.52 0.20 0.04 y 1 0.14 0.02 0.01 2 0.06 0.01 0 1. compute the probability of at least one hardware failure. 2. from the given distribution, are x and y independent? why or why not?

Answer 1

The probability of at least one hardware failure is 0.48. After examining the joint probabilities, it's evident that x and y are not independent because the product of their marginal probabilities does not equal the joint probability for any pair of x and y.

Step-by-step explanation:

To compute the probability of at least one hardware failure in the two computer labs, we need to look at all the scenarios where either one or both labs have failures. This means considering all values of x and y except where both are zero. We can find this probability by adding all the joint probabilities given and subtracting the probability of no failures (x=0 and y=0), which is 0.52. So, the probability of at least one hardware failure is:

1 - P(x=0, y=0) = 1 - 0.52 = 0.48.

To determine whether x and y are independent, we check if the product of the marginal probabilities of x and y equals the joint probability for all x and y. Consider P(x=1) and P(y=1). Their marginal probabilities are obtained by summing the rows and columns, respectively. If x and y were independent, P(x=1)P(y=1) would be equal to P(x=1, y=1). However, we can observe from the given probabilities that this is not the case, hence x and y are not independent.

Therefore, since the product of the marginal probabilities does not equal the joint probability, x and y are not independent.