03-AgeDistributions - AGE DISTRIBUTIONS The Mean Age of an...

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AGE DISTRIBUTIONS The Mean Age of an Individual If we pick an individual at random from a cohort, on average how long will that individual live? The average age attained by an individual in a cohort is also known as the expectation of life and is closely related to the instantaneous rate of mortality. We can demonstrate this by deriving an equation for the average value of t from our equation for survival N t ( ) N 0 ( ) exp M - t ( ) = First, we need to define what we mean by the average of a continuous variable. For a discrete variable, say X, the average value is just _ E(X) = X = ( Σ f i X i ) / ( Σ f i ) where f i is the frequency of individuals in the class i. This is only a little bit different from the normal process of calculating an average. You sum up the measurements and divide by the number of measurements. In this case the measurements have been grouped into categories and tallied. Here is an example. 5 fish 10 cm 50 fish cm = + 13 fish 20 cm 260 fish cm = Notice that the units are homogeneous. + 7 fish 30 cm 210 fish cm = 25 fish 520 fish cm = _ L = ( 520 fish cm ) / ( 25 fish ) = 20.8 cm For a continuous variable, such as t in our problem, the average value is obtained by using integrals rather than summations. Think of the integral as the process of summing up the infinitesimally small slices dY. _ The term f(Y) is called the probability density function for Y. It measures how frequently each value occurs. Y = E Y ( ) Y f Y ( ) Y d Y f Y ( ) d = _ To determine the average age of an individual we need to determine What are the units? t = E t ( ) 0 t N t ( ) t d 0 t N t ( ) d = We have to evaluate the two definite integrals in this ratio. Start by evaluating the indefinite form of the integral in the denominator. FW431/531 Copyright 2008 by David B. Sampson Age Distributions - Page 19
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t N t ( ) d t N 0 ( ) exp M - t ( ) d = N 0 ( ) t exp M - t ( ) d = a·f(X) dX = a· f(X) dX = N 0 ( ) - exp M - t ( ) M C + e aX dX = (1/a)·e aX + C Recall that if we differentiate the solution to the integral, we recover the original integrand . To evaluate the definite form of an integral we take the value of the indefinite integral at the lower limit and subtract it from the value of the indefinite integral at the upper limit . The arbitrary constants cancel when we take the difference to calculate the definite integral. . The lower limit is at t = 0. ==> N 0 ( ) - exp M - 0 ( ) M N 0 ( ) - M = The upper limit is obtained by letting t approach positive infinity. t
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03-AgeDistributions - AGE DISTRIBUTIONS The Mean Age of an...

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