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

Question

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 associated with X? [1 b) What is the probability you find a taxi within five minutes of waiting? [1 (c) What is the probability you have to wait over a half-hour for a taxi? [1 (d) Find the upper quartile of X (i.e. 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 15e1X, where the tip T is measured in dollars. 12l What is the average tip value you will leave (AT)? EC Let Y be the number of taxis that arrive in one hour. What is the distribution of Y?11

Explanation / Answer

a. For exponential distribution, E[X]=1/lambda=10, thus, lambda=0.1

b. P(X<5)=1-e^{(-0.1)*5}=0.3935

c. P(X>30)=1-P(X<=30)=1-[1-e^{(-0.1)*30}]=0.0498

d. From information given, P(X<=x)=0.75, that is 1-e^{(0.1*x)}=0.75; 0.25=e^{0.1*x}, x=-13.86

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