2. (20 points) A company has a call center for the purpose of responding to tele

Question

2. (20 points) A company has a call center for the purpose of responding to telephone inquiries from its customers. A telephone call received from this center is one of either two types: call from an existing customer or call from a new customer. For each hour, the number of calls received from this call center has a Poisson distribution with mean 10 and the percentage of calls from existing customers is 40% Suppose that this same company has 100 call centers all over the world. Each call center are independently run and receives calls from either existing or new customers in exactly the same fashion as described above. (i) (2 points) Describe the distribution of N, the total number of calls received from all 100 call centers on any given hour. (ii) (5 points) Use a suitable approximation to estimate the probability that on a given hour, there are at most 1,100 calls received from all call centers.

Explanation / Answer

(a) Since each of the call centre recieves calls independently

=>

N, the total number of calls received from all 100 call centers also follows a POISSON distribution

with parameter, Lambda = 100*10 = 1000 calls/hour

P(N) = e^(-1000)*(1000)^N / N!

(b)

Since value of Lambda ( = 100) is sufficiently large, it can be approximated as NORMAL distribution

with mean = Lambda = 1000

and std. deviation = sqrt(Lambda) = sqrt(1000) = 10*sqrt(10)

P( N < 1100) , applying continuity correction

= P( X < 1100.5)

= P ( Z < (1100.5 - 1000)/(10sqrt(10)) )

= P ( Z <3.178 )

=0.9993

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