A new product (a \"Home Helicopter\" to solve the commuting problem) is being in

Question

A new product (a "Home Helicopter" to solve the commuting problem) is being introduced. The sales arc expected to be determined by both media (newspaper and television) advertising and word-of-mouth advertising, wherein satisfied customers tell others about the product. Assume that media advertising creates new customers according to a Poisson process of rate alpha = 1 customer per month. For the word-of-mouth advertising, assume that each purchaser of a Home Helicopter will generate sales to new customers at a rate of theta = 2 customers per month. Let X(t) be the total number of Home Helicopter customers up to time t. (a) Model X(t) as a pure birth process by specifying the birth parameters lambda_k for k = 0, 1, .... (b) What is the probability that no Home Helicopters are sold during the first month? (c) What is the probability that exactly one Home Helicopter is sold during the first month?

Explanation / Answer

a.) Lambda for k=0,1,2............ is 3(2+1)

b.) Probability k=0=(exp(-3)*3^0)/0!=0.0497870

c.) probability that k=1=(exp(-3)*3^1)/1!=0.14936

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