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The following problem can be found in van Kampen's "Stochastic Processes in Physics and Chemistry", Third Edition (Exercise I.3.7):

The probability distribution of lifetimes in a population is P(t). Show that the conditional probability for individuals of age τ is

\begin{equation}

P(t|τ) = \frac{P(t)}{\int_τ^{\infty} P(t') dt'} \qquad (t>τ)

\end{equation}

Note that in the case $$P(t)=\gamma e^{-\gamma t}$$ one has $$P(t|\tau)=P(t-\tau),$$ the survival chance is independent of age. Show that this is the only P for which that is true.

Now, I am only interested in the first part of the problem - the expression for P(t|τ). A solution is given here:

http://www.claudiug.com/9780444529657/chapter.php?c=1&e=38

and I find it correct.

What I do not understand is:P(t|τ) is a conditional probability density. As such it should be expressed as a fraction of two probability densities. However, the denominator in the problem above is NOT a probability density, but actual probability (that the particle has age >τ).

Where is the catch?

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# Why is this correct - particle lifetime probability distribution

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