Question

Applies Stats for Engineers and Scientists

A mail-order computer business has six telephone lines. Let x denote the number of lines in use at a specified time. Suppose the mass function of x is given by 1 5 6 2 3 4 0 x: p(x): .10 .15 20 .25 .20?? a. In the long run, what proportion of the time will at most three lines be in use? Fewer than three lines? b. In the long run, what proportion of the time will at least five lines be in use? c. In the long run, what proportion of the time will between two and four lines, inclusive, be in use? d. In the long run, what proportion of the time will at least four lines not be in use?

Answer 1

Solution(a)

At most 3 lines

P(X<=3) = P(X=0) + P(X=1) +P(X=2) +P(X=3) = 0.10+0.15+0.2+0.25 = 0.7

Fewer than 3

P(X<3) = P(X=0) + P(X=1) + P(X=2) = 0.1+0.15+0.2 = 0.45

Solution(b)

P(X>=5) = 1- P(X=0) - P(X=1) -P(X=2) - P(X=3) - P(X=4) = 1-0.1-0.15-0.20-0.25-0.20 = 0.1

Solution(c)

P(2<=X<=4) = P(X=2)+ P(X=3)+P(X=4) = 0.20+0.25+0.20= 0.65

Solution(d)

Atleast 4 lines in use

P(X>=4) = 1- P(X=0)-P(X=1)-P(X=2)-P(X=3)= 1-0.10-0.15-0.20-0.25-= 0.3

Atleast 4 lines not in use

1-P(X>=4) = 1-0.3= 0.7