- #1

Quireno

- 11

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## Homework Statement

Let P(t) be a population at time t, b(t) is the birth rate at time t and d(t) is the death rate at time t.

- Define the relative mortality μ(t) as the probability to die at age t.
- How is it normalized?
- Set up an equation of the absolute number of deaths at time t as a convolution of b(t) and μ(t).
- Suppose the relative mortality decays exponentially as a function of age [tex]\mu(t)=\frac{e^{-t/T}}{T}[/tex] Is it correctly normalized?

## Homework Equations

[tex]P(t)=P_0 +b(t)\cdot t-d(t)\cdot t[/tex]

[tex]\frac{d}{dt}P(t)=b(t)-d(t)[/tex]

## The Attempt at a Solution

- The first part was confusing because of the ambiguity age=time but I figured out that if you take into acount all ages you can define an
__absolute__mortality at a time t [tex]μ(t)=d(t)/P(t)[/tex] but I can be wrong. - This part was confusing as well. The only help I could get was: "The mortality is normalized taking into acount that everyone must die at a certain age" so my attempt was [tex]P_0=\Sigma\mu(t)P(t)[/tex] because the total population at a time is the sum of the people who dies from that moment (b=0, of course) but I'm not so sure about this. Also I'd want to know if the sums can be converted into an integral so I can plug it into one of the equations I got.
- So it basically asks me to do something like this: [tex]d(t)=(μ*b)(t)=

\int_{0}^{\infty}\mu(t-t')b(t')dt'[/tex] and then replace into equation in 2. and solve the equation using Laplace transforms... but I can't see why the appliance of convolution is valid in this case. - Ignoring the fact that it doesn't make sense that mortality decays with time, I know that [tex]\int\frac{e^{-t/T}}{T}dt=e^{-t/T}[/tex] but I think it is not correctly normalized because it should be equal to 1 but it does not seems correct divide the thing by ##e^{-t/T}## am I getting something wrong?

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