Explanation Given: f ( t ) = e t 10 function represents the fraction of people who will retain their membership in a computer dating service for “t” months. The number of charter member is equal to 8 , 000 Calculation: Survival function: it is function that gives the fraction of individuals within a group of population. This fraction value is expected to remain in the group of that population. Renewal function: it is rate, at which new members can be expected to join the group of population. Such survival and renewal function problems occur in many fields like ecology, sociology, demography and finance. It results the investment strategy. For example, f ( t ) = e t 10 function represents the fraction of people who will retain their membership in a computer dating service for “t” months. The number of charter member is equal to 8 , 000 So, out of the 8 , 000 members, those who are retain their membership after 10 months will be, 8 , 000 f ( 10 ) = 8 , 000 e 10 10 = 8 , 000 e − 1 The new members attracted at the rate of 200 per month. Nearly, the number of new members will be active after 10 months, divide the interval 0 ≤ t ≤ 10 into “n” equals to subintervals of length of 𑩇 t months. Suppose, assume that t j − 1 represents the beginning of j t h sub-interval. So the number of new members accepted during the j t h subinterval will be 200 𑩇 t . From now, after 10 months, for the j t h sub-interval, the remaining months will be 10 − t j − 1 .So out of the 200 𑩇 t members; approx. 200 𑩇 t . f ( 10 − t j − 1 ) members will retain their membership at that time. Therefore, the total number of new members will retain their membership can be approximated by the sum ∑ j = 1 n 200 𑩇 t . f ( 10 − t j − 1 ) Adding the current members who will retain their membership after 10 months, so, the total number of members who will retain their membership after 10 months from now will be, P = 8 , 000 e − 1 + ∑ j = 1 n 200 𑩇 t . f ( 10 − t j − 1 ) As the “n” increases without bound, the approximation improves and approaches the true value of “p”. P = 8 , 000 e − 1 + lim n → ∞ ∑ j = 1 n 200 𑩇 t

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Chapter 6.6, Problem 4PS

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Ch. 6.1 - Prob. 1PS Ch. 6.1 - Prob. 2PS Ch. 6.1 - Prob. 3PS Ch. 6.1 - Prob. 4PS Ch. 6.1 - Prob. 5PS Ch. 6.1 - Prob. 6PS Ch. 6.1 - Prob. 7PS Ch. 6.1 - Prob. 8PS Ch. 6.1 - Prob. 9PS Ch. 6.1 - Prob. 10PS

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the survival function and the renewal function meaning

Solution Summary: The author calculates survival function and renewal function. Survival function gives the tion of individuals within a group of population. Renewal function is rate, at which new members can be expected to join.

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