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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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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 Fall '09
 Demography, Derivative, Frequency distribution

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