Exponential density 3

In example 4.20 of Grinstead and Snell there is a nice conjunction of Bayes theorem and use of the exponential density. Recall that the exponential pdf is:



To find the probability that X happens (hard-drive failure, radioactive decay) within a certain time period, we integrate the pdf over the interval. For example, the probability that the failure happens after a particular time t is:



(Naturally, since the cdf(t) is 1 minus this value).

Here is one form of Bayes theorem:



Now, consider two events E and F defined as follows:

E is the event that failure happens after time r F is the event that failure happens after time r + s

Note that P(F and E) = P(F) because F is totally contained within E.

Then:



The probability of failure after time r + s, when we know that failure occurs after time r, does not depend on r at all but is only a function of s. This is the memoryless property of the exponential function, alluded to in a previous post.