Respuesta :
Answer:
[tex]P(X < 3) = 67.14\%[/tex]
Therefore, 67.14% of the cars can get through the toll booth in less than 3 minutes.
Explanation:
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.
Let X be a random variable. The service time has an exponential distribution with a mean of 2.7 minutes.
Mean = μ = 2.7 minutes
Decay rate = m = 1 /μ
Decay rate = m = 1/2.7
Decay rate = m = 0.371
The cumulative probability distribution function is given by
[tex]P(X < x) = 1 - e^{-mx}[/tex]
Where m is the decay rate and x < 3
So the proportion of cars that can get through the toll booth in less than 3 minutes is
[tex]P(X < 3) = 1 - e^{-0.371(3)}[/tex]
[tex]P(X < 3) = 1 - 0.3286[/tex]
[tex]P(X < 3) = 0.6714[/tex]
[tex]P(X < 3) = 67.14\%[/tex]
Therefore, 67.14% of the cars can get through the toll booth in less than 3 minutes.
