
Posterior distribution
Let theta denote the average number of defects per 100 feet of a certain type of magnetic tape. suppose that the value of the mean theta is unknown and the prior distribution of theta is a gamma distribution with parameters alpha=2 and beta=10. when a 1200 foot roll of this tape is inspected, exactly four defects are found. determine the posterior distribution of theta.
i'm pretty sure the posterior is a gamma function i just need to figure out alpha and beta. do i multiply by 12...?

It seems to me that the distribution of the number of defects per 100 feet is missing. I'm guessing they want you to think of it as a Poisson process. If X is the number of defects in the 1200 foot thing, then we should have Xtheta ~ Poisson(12theta). So, you want to get the distribution of thetaX, evaluated at X = 4. Use the fact that
$\displaystyle f_{\thetaX}(\thetax) \propto f_{X\theta}(x\theta) f_\theta(\theta)$.
You should get a gamma for the posterior I think.

i got a gamma with alpha= 6 and beta= 22 for the posterior is this right??

If you are using the rate parameterization of the gamma (which is NOT the one given here), then yes.

what do you mean by the rate parameterization of the gama?

It is equivalent to using paramaterizing using lambda = 1 / beta in the paremeterization given on wikipedia.