Why is this correct - particle lifetime probability distribution

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Hello,

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?
 

Answers and Replies

  • #2
DrDu
Science Advisor
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/B] P(t|τ) is a conditional probability density. As such it should be expressed as a fraction of two probability densities.
no, why?
After all, you condition on a real probability, namely that t>τ, not on a density.
 
  • #3
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Exactly, right?
But the definition says otherwise.

My point is: from the definition of probability density,
$$P_X(x) dx$$
is the probability that a random variable X has a value in [x,x+dx]. P(x) by itself is NOT the probability that the value of the random variable X will by precisely x. That is clear.

But the conditional probability density P_{X|Y}(x|y) has the following interpretation:
$$ P_{X|Y}(x|y) dx $$ is the probability that X will have a value from the set [x,x+dx], given that Y==y. Precisely equal, and not that Y will be in [y,y+dy].

And the Bayes' rule for conditional probability density is:
$$
P_{X|Y}(x|y) = \frac{P_{X,Y}(x,y)}{P_Y(y)}
$$
where all P-s, in particular P_Y, are probability densities.
(If the denominator vanishes, the conditional probability is not defined - conditions cannot be met.)
P_Y(y) is not the probability that Y==y.

And yet (going back to my original problem) I have P(τ) in the denominator, which is not a probability density, but a crude probability.

Why isn't there a probability density, that the lifetime of the particle is τ?
 

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