An exponential probability distribution has lambda equal to 16 customers per hou

Question

An exponential probability distribution has lambda equal to 16 customers per hour. Find the following probabilities. a) What is the probability that the next customer will arrive within the next 3 minutes? b) What is the probability that the next customer will arrive within the next 45 seconds? c) What is the probability that the next customer will arrive within the next 7 minutes? d) What is the probability that the next customer will arrive within the next 17 minutes? a) The probability that the next customer will arrive within the next 3 minutes is. b) The probability that the next customer will arrive within the next 45 seconds is. c) The probability that the next customer will arrive within the next 17 minutes is. d) The probability that the next customer will arrive within the next 17 minutes is .

Explanation / Answer

Solution: We are given that , random variable t = time between next customer will arrive follows an exponential distribution with rate per hour = = 16 customer hour = 16 customer per 60 minutes.

Hence rate per minute = = 16 / 60   ( we use this conversion, since all parts are asked in Minutes)

Part a) We have to find : The probability that next customer will arrive within the next 3 minutes

That is , we have to find : P( T 3 ) =.................?

Distribution function for Exponential distribution with rate parameter is given by :

                Ft(T) = P( T t ) = 1 - e -t

   Thus , P( T 3 ) = 1 - e -3 * (16/60)

                                = 1 - e - 48/60

                                = 1 - e - 0.8

                                = 1 - 0.4493

         P( T 3 )       = 0.5507

Thus , the probability that next customer will arrive within the next 3 minutes is 0.5507 .

Part b) We have to find : The probability that next customer will arrive within the next 45 minutes

That is , we have to find : P( T 45 ) =.................?

Thus , P( T 45 ) = 1 - e -45 * (16/60)

                                = 1 - e - 720 /60

                                = 1 - e - 12

                                = 1 - 0.000006

         P( T 45 )       = 0.999994 = 1.0000

Thus , the probability that next customer will arrive within the next 45 minutes is 1.0000.

Part c) We have to find : The probability that next customer will arrive within the next 7 minutes

That is , we have to find : P( T 7 ) =.................?

Thus , P( T 7 ) = 1 - e -7 * (16/60)

                                = 1 - e - 112/60

                                = 1 - e - 1.8667

                                = 1 - 0.1546

         P( T 7 )       = 0.8454

Thus , the probability that next customer will arrive within the next 7 minutes is 0.8454.

Part d) We have to find : The probability that next customer will arrive within the next 17 minutes

That is , we have to find : P( T 17 ) =.................?

Thus , P( T 17 ) = 1 - e -17 * (16/60)

                                = 1 - e - 272 /60

                                = 1 - e - 4.5333

                                = 1 - 0.0107

         P( T 17 )       = 0.9893

Thus , the probability that next customer will arrive within the next 17 minutes is 0.9893.

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