9/18 I forgot my math. Say the probability of a bug happening is the
unknown fixed value p in [0,1]. I attempt to reproduce the
bug until it happens once. Then it happens at my n-th trial, so I
stop. Now, what is the expected value of p? Is it E(p) = 1/n? Thx.
\_ Did a quick program to test. Looks like E(p) = 1/n
(given assumption n(p) = sum x = 1 to inf of x*(1-p)^(x-1)*p
where n(p) = expected n for a given p)
\_ Consider the case n=1 to see why E(p) = 1/n can't possibly be
right. You need to specify a prior distribution and then apply
Bayes' theorem. If your prior distribution is the uniform
distribution on [0,1] (i.e. before running the experiment you think
all values for p between 0 and 1 are equally likely), then if you
see your bug for the first time in the nth trial, you'll get a
posterior distribution of (n)(n+1)(p)(1-p)^(n-1). The expected
value of this is the root r of the equation (1+nr)(1-r)^n = 1/2.
For n = 1, E(p)=1/sqrt(2). For n = 2, E(p) = 1/2. For n = 3, it's
around .3857. This sequence probably has some name, but I don't
know it. [In practice that might be a stupid prior to use for your
problem. But you have to have some prior, otherwise the problem is
not well-specified mathematically.]
\_ Doy, wow I'm an idiot. I'm wrong, this person is right. -pp
\_ That is a fine answer above. The part about Bayes Rule and
having to make some assumptions about priors/distribution
properties is hard to explain. All I have to add is OP may
enjoy reading about the NEGATIVE BINOMIAL DISTRIBUTIION.
(while you are at it, you might read up on POISSON DISTRIBUTION/
PROCESS ... which can be quite useful in a lot of engineering
contexts). |