4·Let X be the amount of time (measured in minutes) you have to wait for the arr

Question

4·Let X be the amount of time (measured in minutes) you have to wait for the arrival of a taxi at some particular location:; assume that X has the exponential distribution with a mean of 10 minutes (a) What is the rate parameter asociated with X? (b) What is the probability you find a taxi within five minutes of waiting? (c) What is the probability you have to wait over a half-hour for a taxi? (d) Find the upper quartile of X (ie. the 0.75-quantile of X), and interpret this number.3 (e) Suppose that your generosity wanes as you wait longer for a cab, so that the amount you tip the driver is described by T = 15e-01x, where the tip T is measured in dollars. 2) What is the average tip value you will leave (T)? EC Let Y be the number of taxis that arrive in one hour. What is the distribution of Y1]

Explanation / Answer

Q.4

(a) Here Rate parameter = 1/10 = 0.1 min-1

f(X) = 0.1e-0.1x ; 0 < x

F(x) = 1 - e-0.1x

(b) Pr (X < 5 minutes ) = EXP(X < 5 minutes) = 1 - e-0.1 * 5 = 1 - e-0.5  = 0.3935

(c) Pr (X > 30 minutes ) =EXP(X > 30 minutes) = 1 - [1 - e-0.1 * 30]  = e-3 = 0.0497

(d) Upper quartile of X

F(X) = 0.75

1 - e-0.1X = 0.75

e-0.1X = 0.25

-0.1 X = -1.3863

X= 13.863 minutes

so there is 75% chance that the passenger have to wait less than 13.86 minutes

(e) here Tip T = 15 e-0.1X

so Expected Tip E(T) = E(15 e-0.1X ) = E(150 * 0.1 e-0.1X)  = 150 * E(0.1 e-0.1X) = 150 * (1/10) = $ 15

(f) Here Y is the number of taxies arrive in one hour that will be a possion distribution with = 6

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