# GE330practice5 - Practice 5 Pure Birth Pure Death...

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Practice 5: Pure Birth, Pure Death, Generalized Poisson Model April 28, 2009 Example 15.4, Exercise 1 on page 559. PURE BIRTH Babies are born at the rate of one birth every 12 minutes. The time between births has exponential distribution. Find the following: (a) The average number of births per year. (b) The probability that n births will occur in any one day. (c) The probability of issuing 50 birth certiﬁcates in 3 hours given that 40 certiﬁ- cates were issued during the ﬁrst 2 hours. (d) Suppose that a clerk enters the information from birth certiﬁcates into a com- puter whenever 5 certiﬁcates have accumulated. What is the probability that the clerk will be entering a new batch every hour? Solution: Recall, the number of arrivals (at rate λ ) during a speciﬁed period of time T has the following properties: (1) The expected number of arrivals is λT . (2) The probability of n arrivals ( n 0 integer) in period T is p n ( T ) = ( λT ) n e - λT n ! . First, we ﬁnd the birth rate per day: λ = 1 12 × 60 × 24 = 120 births per day . (a) The expected number of births per year is λt = 120 × 365 = 43 , 800 births per year . (b) This is p n ( T ) where T is a day and n = 0, so we have p 0 (1) = (120 × 1) 0 e - 120 × 1 0! = e - 120 0 . 1

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(c) This is the same as the probability of issuing 10 certiﬁcates in an hour. The rate is 1 certiﬁcate per 12 minutes or 5 certiﬁcates per hour. So the probability is p n ( T ) for n = 10 and T = 1 hour, i.e., p 10 (1) = (5 × 1) 10 e - 5 × 1
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GE330practice5 - Practice 5 Pure Birth Pure Death...

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