Question

Toll booths on the New York State Thruway are often congested because of the large number of cars waiting to pay. A consultant working for the state concluded that if service times are measured from the time a car stops in line until it leaves, service times are exponentially distributed with a mean of 2.7 minutes. What proportion of cars can get through the toll booth in less than 3 minutes?

Answer 1

Answer:

The proportion of cars can get through the toll booth in less than 3 minutes is 67%.

Step-by-step explanation:

Let the random variable X be defined as the service times at a tool booth.

The random variable X follows an Exponential distribution with parameter μ = 2.7 minutes.

The probability density function of X is:

$f_{X}(x)=\frac{1}{\mu}e^{-x/\mu };\ x\geq 0,\ \mu>0$

Compute the probability that a car can get through the toll booth in less than 3 minutes as follows:

$P(X<3)=\int\limits^{3}_{0} {\frac{1}{2.7}\cdot e^{-2.7x}} \, dx$

                $=\frac{1}{2.7}\cdot \int\limits^{3}_{0} {e^{-x/2.7}} \, dx \\\\=\frac{1}{2.7}\cdot [-\frac{e^{-x/2.7}}{1/2.7}]^{3}_{0}\\\\=1-e^{-3/2.7}\\\\=0.6708$

Thus, the proportion of cars can get through the toll booth in less than 3 minutes is 67%.