People Power, Inc., is a firm that specializes in providing temporary help to various businesses. Job applicants are administered an aptitude test that evaluates mathematics, writing, and manual dexterity skills. After the firm analyzed thousands of job applicants who took the test, it was found that the scores on the three tests could be viewed as outcomes of random variables with the following joint density function (the tests are graded on a \(0-1\) scale, with 0 the lowest score and 1 the highest):

\(f\left(x_{1}, x_{2}, x_{3}ight)=.80\left(2 x_{1}+3 x_{2}ight) x_{3} \prod_{i=1}^{3} I_{[0,1]}\left(x_{i}ight)\).

a. A job opening has occurred for an office manager. People Power, Inc., requires scores of \(>.75\) on both the mathematics and writing tests for a job applicant to be offered the position. Define the marginal density function for the mathematics and writing scores. Use it to define a probability space in which probability questions concerning events for the mathematics and writing scores can be answered. What is the probability that a job applicant who has just entered the office to take the test will qualify for the office manager position?

b. A job opening has occurred for a warehouse worker. People Power, Inc., requires a score of \(>.80\) on the manual dexterity test for a job applicant to be offered the position. Define the marginal density function for the dexterity score. Use it to define a probability space in which probability questions concerning events for the dexterity score can be answered. What is the probability that a job applicant who has just entered the office to take the test will qualify for the warehouse worker position?

c. Find the conditional density of the writing test score, given that the job applicant achieves a score of \(>.75\) on the mathematics test. Given that the job applicant scores \(>.75\) on the mathematics test, what is the probability that she scores \(>.75\) on the writing test? Are the two test scores independent random variables?

d. Is the manual dexterity score independent of the writing and mathematics scores? Why or why not?