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## Homework Statement

Consider a population of individuals with a disease. Suppose that [tex]t[/tex] is the number of years since the onset of the disease. The death density function, [tex]f(t) = cte^{-kt}[/tex], approximates the fraction of the sick individuals who die in the time interval [t, t+Δt] as follows:

Fraction who die: [tex]f(t)\Delta t = cte^{-kt} \Delta t[/tex]

where [tex]c[/tex] and [tex]k[/tex] are positive constants whose values depend on the particular disease.

(a) Find the value of c in terms of k.

(b) Express the cumulative death distribution function in the form below. Your answer will be in terms of k.

[tex]

C(t)=\left\{\begin{array}{cc}A(t),& t < 0\\

B(t), & t \geq 0\end{array}\right

[/tex]

## Homework Equations

[tex]P(t) = \int_{-\inf}^t p(x) dx[/tex]

## The Attempt at a Solution

To solve part a, I know that [tex]\lim_{t\rightarrow\infty} P(t) = 1[/tex]. So c and k must have values so that it equals one. So I integrate [tex]f(t)[/tex] with the relevant equations to get:

[tex]

\frac{-(kt+1) \cdot e^{-kt} \cdot c}{k^2}

[/tex]

from negative infinity to t.

The problem is that this diverges to negative infinity and doesn't give me a meaningful answer. So what do I do?

Also, how do I get the the bar with a superscript and subscript for 'from a to b' in tex?