A small nursery has seven employees, three of whom are salespersons, and four of whom are gardeners who tend to the growing and caring of the nursery stock.

With such a small staff, employee absenteeism can be critical. The number of salespersons and gardeners absent on any given day is the outcome of a bivariate random variable \((X, Y)\). The nonzero values of the joint density function are given in tabular form as:

a. What is the probability that more than two employees will be absent on any given day?

b. Find the marginal density function of the number of gardeners that are absent. What is the probability that more than two gardeners will be absent on any given day?

c. Are the number of gardener absences and the number of salesperson absences independent random variables?

d. Find the conditional density function for the number of salespersons that are absent, given that there are no gardeners absent. What is the probability that there are no salespersons absent, given that there are no gardeners absent? Is the conditional probability higher or lower given that there is at least one gardener absent?

Transcribed Image Text:

0 0 .75 X 1 .06 2 .025 3 .005 Y 1 3 4 025 .01 .03 .03 .01 .003 ,01 .005 .005 .002 .004 .003 .002 .001 2 .01 .01