I. In a mid-size city in Texas, 21% of all cars on the road are made by Ford. Th
ID: 3305592 • Letter: I
Question
I. In a mid-size city in Texas, 21% of all cars on the road are made by Ford. The Ford corporation is doing market research in the town and has a representative of theirs wait at the intersection of Main and State observes the first Ford. Let Y = the number of cars that go by until the first Ford is observed. (a) What is the distribution of Y? Write down the pmf. (b) Calculate E (Y) and SD (Y). (c) What is the probability that the rep observes (i) between 3 and 5 cars (inclusive)? (ii) more than 5 cars?Explanation / Answer
a. If one is interested in determining how long it takes to achieve the first success in a Bernoulli trials, the model that tells the probability is called geometric probability model. The model is specified by one parameter, probability of success, p. Since, achieving the first success in trial number x requires first experienceing x-1 failures, the probabilities are easily calculated by using P(X=x)=q^(x-1)p
The pmf is as follows:
Y=0 P(y=0)=0.79^(-1)*0.21=0.2658
Y=1 P(y=1)=0.79^0*0.21=0.2100
b. E[Y]=mu=1/p=1/0.21=4.7619
SD[Y]=sqrt [q/p^2]=sqrt [0.79/0.21^2]=4.2325
c. i) P[3<=Y<=5]=P[Y=3]+P[Y=4]+P[Y=5]=0.79^2*0.21+0.79^3*0.21+0.79^4*0.21=0.3164
ii) P(Y>5)=1-P[Y<=5]=1-[0.79^0*0.21+0.79^1*0.21+0.79^2*0.21+0.79^3*0.21+0.79^4*0.21]=0.3077