Explanation Calculation: Survival and Renewal: Survival function is a function that gives the fraction of individuals in a group of the population that can be expected to remain in the group. A renewal function gives the rate at which new members can be expected to join the group. Survival and renewal problems arise in many fields including Sociology, Ecology, Demography and Finance. It refers to results of an investment strategy. A typical survival and renewal problem is as follows: “The operators of a new telecom service estimates that the fraction of people who will retain their membership for at least t months is given by the function f ( t ) = e t 20 There are 5000 current members and the operators expect to attract 120 members per month. How many members will the service have 9 months from now.” The function that represents the fraction of people who will retain their membership in the service for at least t months is, f ( t ) = e t 20 …………… (1) Out of current 5.000 members, number of members who will retain their membership after 9 months will be, 5 , 000 f ( 9 ) = 5 , 000 e 9 20 New members will be attracted at the rate of 120 per month. Approximate the number of new members who will be active after 9 months, divide the interval 0 ≤ t ≤ 9 into n equal subintervals of length Δ t months. Assume that t j − 1 represents the beginning of j t h sub-interval, so number of new members accepted during the j t h sub-interval will be 120 Δ t . 9 months from now, for the j t h sub-interval, remaining months will be 9 − t j − 1 . So, out of these 120 Δ t new members, approximately 120 Δ t ⋅ f ( 9 − t j − 1 ) members will retain their membership at that time. So, the total number of new members who will retain their membership 9 months from now can be approximated by the sum ∑ j − 1 n 120 f ( 9 − t j − 1 ) ⋅ Δ t Add the number of current members who will retain their membership after 9 months, total number of members who will retain their membership 9 months from now will be, p = 5 , 000 e 9 20 + ∑ j − 1 n 120 f ( 9 − t j − 1 ) ⋅ Δ t As n increases without bound, the approximation improves and approaches the true value of P

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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A typical survival and renewal problem

Solution Summary: The author explains survival and renewal problems, which arise in many fields including sociology, ecology, demography, and finance.

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