Question

minutes. What is the probability that you wait longer than one hour for a taxi? (i) (ii) one arrives within the next 10 minutes? Determine x such that the probability that you wait more than x minutes is 0.10. (iii)

Answer 1

Solution :

Let X be a random variable which represents the time between arrivals of taxis at a busy intersection.

Given that, X is exponentially distributed with mean mean of 10 minutes.

The probability density function of a exponentially distributed random variable X with mean of 10 minutes is given as follows :

x > 0

i) We have to obtain P(X > 1 hours).

P(X > 1 hours) = P(X > 60 minutes).

Hence, the probability that you wait longer than one hour is 0.0025.

(ii) I have been already been waiting for one hour for a taxi and I need to obtain that a taxi will arrive within next 10 minutes.

It means we need to obtain the a taxi will arrive between 60 and 70 minutes i.e. we need to obtain P(60 < X < 70).

Hence, the probability that one arrives within next 10 minutes is 0.0016.

(iii) To determine x such that P(X > x) = 0.10

Taking natural logarithm both side we get,

The value of x is 23.026.

(iv) To determine x such that P(X < x) = 0.90

Taking natural logarithm both side we get,

The value of x is 23.026.