Question

The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 10 minutes. (a) What is the probability that you wait longer than one hour for a taxi? (b) Suppose that you have already been waiting for one hour for a taxi. What is the probability that one arrives within the next 10 minutes? (c) Determine x such that the probability that you wait more than x minutes is 0.10. (d) Determine x such that the probability that you wait less than x minutes is 0.90.

Answer 1

Answer:

a) 0.0025

b) 0.9975

c) 23.03 minutes

d) 23.03 minutes

Step-by-step explanation:

Let X be the random variable that measures the time waited for a taxi.  

If X is exponentially distributed with a mean of 10 minutes,then the probability that you have to wait more than t minutes is  

$\large P(X>t) = e^{-t/10}$

a)  

1 hour = 60 minutes, so the probability that you wait longer than one hour is

$\large P(X>60)=e^{-6}=0.0025$

b)

Due to the “memorylessness” of the exponential distribution, the probability that you have to wait 10 or less minutes after you have already waited for one hour, is the same as the probability that you have to wait 10 or less minutes

$\large P(X\leq 10)=1-e^{-6}=0.9975$

c)

We want x so that

P(X>x)=0.1

$\large e^{-x/10}=0.1\rightarrow -x/10=ln(0.1)\rightarrow x=-10ln(0.1)=23.03\;minutes$

d)

We want P(X<x)=0.9

$\large 1-e^{-x/10}=0.9\rightarrow e^{-x/10}=0.1\rightarrow x=-10ln(0.1)=23.03\;minutes$