An automobile service center can take care of 8 cars per day. Assume that the ca

Question

An automobile service center can take care of 8 cars per day. Assume that the cars arrive at the service center randomly and independently of each other at a rate of 6 per hour, on average. What is the standard deviation of the number of cars that arrive at the center? ______________ What is the probability of the service center being empty in any given hour? ______________ What is the probability that exactly 6 cars will be in the service center at any point during a given hour? ______________ What is the probability that less than 2 cars will be in the service center at any point during a given hour? _____________

Explanation / Answer

Let X be the random variable that number of cars will be in the service center at any point during a given hour.

X ~ Poisson( = 6)

Then X has probability mass function is,

P(X=x) = (e- * x ) / x!

What is the standard deviation of the number of cars that arrive at the center?

standard deviation = sqrt( ) = sqrt(6) = 2.449

What is the probability of the service center being empty in any given hour?

That is here we have to find P(X=0).

P(X=0) = (e-6 * 60) / 0! = 0.0025

What is the probability that exactly 6 cars will be in the service center at any point during a given hour?

That is here we have to find P(X=6).

P(X=6) = (e-6 * 66) / 6! = 0.1606

What is the probability that less than 2 cars will be in the service center at any point during a given hour?

That is here we have to find P(X<2).

P(X < 2) = P(X=0) + P(X=1)

= 0.0025 + (e-6 * 61) / 1!

= 0.0025 + 0.0149

= 0.0174

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