There are two traffic lights on a commuter's route to and fromwork. Let X1 be the number of lights at whichthe commuter must stop on his way to work, andX2 be the number of lights at which he muststop when returning from work. Suppose that these two variables areindependent, each with the pmf given in the accompanying table (soX1, X2 is a random sampleof size n = 2).
| x1 | 0 | 1 | 2 | | μ = 1.2,σ2 = 0.76 |
| p(x1)Â Â | Â Â 0.3Â Â | Â Â 0.2Â Â | Â Â 0.5Â Â |
(a) Determine the pmf of To =X1 + X2.
| to | 0 | 1 | 2 | 3 | 4 |
| p(to) | Â Â Â Â | Â Â Â Â | Â Â Â Â | Â Â Â Â | Â Â Â Â |
(b) CalculateμTo.
μTo =
How does it relate to μ, the population mean?
μTo =  ·μ
(c) CalculateσTo2.
How does it relate to σ2, the populationvariance?
σTo2=  · σ2
(d) Let X3 and X4 be thenumber of lights at which a stop is required when driving to andfrom work on a second day assumed independent of the first day.With To = the sum of all fourXi's, what now are the values ofE(To) andV(To)?
(e) Referring back to (d), what are the values of
P(To= 8) andP(To≥ 7)
[Hint: Don't even think of listing all possibleoutcomes!] (Enter your answers to four decimal places.)
