A manufacturing facility needs to open a new assembly line in four months or there will be significant cost overruns. The manager of this project believes that there are four possible values for the random variable X (the number of months from now it will take to complete this project): 2.5, 3, 4, and 5. It is currently believed that the probabilities of these four possibilities are in the ratio 1 to 2 to 2 to 3. That is, X = 3 is twice as likely as X = 2.5 and X = 5 is 1.5 times as likely as X = 4. Answer the following questions.

(1) Find the probability distribution of X. (Use appropriate mathematical expressions wherever you can to answer this question.) (3 points)

(2) What is the probability that this project will be completed in at least 3 months from now? (2 points)

(3) What is the probability that this project will not be completed on time? (2 points)

(4) What is the expected completion time (in months) from now for this project? (3 points)

(5) How much variability (in months) exists around the expected value found in question (4)? Give both the variance and standard deviation. (3 points)