Question

Scenario: There is only one teller working at a bank. The teller
takes an average of 3 minutes to service a customer. Assume that
the time the teller takes to service a customer can be represented
as an Exponential Probability distribution.

Customers arrive at the teller line at the average rate of 1every
10 minutes. Their arrival pattern follows a Poisson
distribution.

Of 200 customers who get serviced by the teller, how many can
expect to be serviced in more than 6 minutes?:

Answer #1

Given :

Teller takes an average of 3 minutes to service a customer.

Let x be the number of customers taken by teller.

x follow exponential distribution

Therefore λ = 1 / 3

Cumulative distribution function of exponential distribution :
P( X ≤ x ) = 1 - e^{-λx}

We have to find P( x> 6 ) = 1 - P( x ≤ 6 )

P( x ≤ 6 ) = 1 - e^{-(6/3) } = 1 -
e^{-2} = 0.8647

P( x> 6 ) = 1 - P( x ≤ 6 ) = 1 - 0.8647

**P( x > 6) = 0.1353**

Therefore number of customers expect to be serviced in more than 6 minutes = 200*P( x > 6) = 200*0.1353 =27.06 ~ 27

Therefore **approximately 27** customers are expect
to be serviced in more than 6 minutes

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