The call center of a medium bank has been receiving phone calls at an average ra

Question

The call center of a medium bank has been receiving phone calls at an average rate of 180 calls per hour.

How many telephone calls should the call center expect in a 5-minute period? Please show how you arrived at your answer.

Define X to be the number of telephone calls to the call center in a 5-minute period and assume that X has a Poisson Probability distribution. Please calculate the probability that the call center will receive 20 calls in the 5-minute period.

Please calculate the probability that the call center will receive at most 12 calls in the 5-minute period.

Please calculate the probability that the call center will receive more than 15 calls in the 5-minute period. Show your work.   

Please calculate the probability that the call center will receive at least 10 calls in the 5-minute period. Show your work.   

Please calculate the probability that the call center will receive fewer than 12 calls in the 10-minute period. Show your work.

Calculate and interpret the standard deviation of X.   

For what purpose can the bank use the kind of information you have calculated above? Please explain.                                                        

Explanation / Answer

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials
a.
The call center of a medium bank has been receiving phone calls at an average rate of 180 calls per hour.
for 5 minuete period calls at average period follows possion distribution
and the rate of interval is = 180 * 5 / 60 = 15 calls per 5 minuete
b.
calculate the probability that the call center will receive 20 calls in the 5-minute period
P( X = 20 ) = e ^-15 * 15^20 / 20! = 0.0418
c.
P( X < = 15) = P(X=15) + P(X=14) + P(X=13) + P(X=12) + P(X=11) + ...... + P(X=0)
= e^-15 * 15 ^ 15 / 15! + e^-15 * 20 ^ 14 / 14! + e^-15 * ^ 13 / 13! + e^-15 * ^ 12 / 12! + e^-15 * ^ 11 / 11! + .....
.. + e ^-15 * 15^0 / 0! = 0.5681
P( X > 15) = 1 -P ( X <= 15) = 1 - 0.5681 = 0.4319
d.
P( X < 10) = P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= e^-15 * 0 ^ 9 / 9! + e^-15 * ^ 8 / 8! + e^-15 * ^ 7 / 7! + e^-15 * ^ 6 / 6! + e^-15 * ^ 5 / 5! + e^-15 * ^ 4 / 4! + e^-15 * ^ 3 / 3! + e^-15 * ^ 2 / 2! + e^-15 * ^ 1 / 1! + e^-15 * ^ 0 / 0!
= 0.0699
P( X > = 10 ) = 1 - P (X < 10) = 0.9301
e.
P( X < 12) = P(X=11) + P(X=10) + P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= e^-15 * 0 ^ 11 / 11! + e^-15 * ^ 10 / 10! + e^-15 * ^ 9 / 9! + e^-15 * ^ 8 / 8! + e^-15 * ^ 7 / 7! + e^-15 * ^ 6 / 6! + e^-15 * ^ 5 / 5! + e^-15 * ^ 4 / 4! + e^-15 * ^ 3 / 3! + e^-15 * ^ 2 / 2! + e^-15 * ^ 2 / 2! + e^-15 * ^ 1 / 1! + e^-15 * ^ 0 / 0!
= 0.1848

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