# Posterior distribution

Printable View

• Feb 8th 2011, 07:09 PM
holly123
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...?
• Feb 8th 2011, 08:16 PM
theodds
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 X|theta ~ Poisson(12theta). So, you want to get the distribution of theta|X, evaluated at X = 4. Use the fact that

$f_{\theta|X}(\theta|x) \propto f_{X|\theta}(x|\theta) f_\theta(\theta)$.

You should get a gamma for the posterior I think.
• Feb 8th 2011, 09:46 PM
holly123
i got a gamma with alpha= 6 and beta= 22 for the posterior is this right??
• Feb 9th 2011, 07:16 AM
theodds
If you are using the rate parameterization of the gamma (which is NOT the one given here), then yes.
• Feb 9th 2011, 07:46 AM
holly123
what do you mean by the rate parameterization of the gama?
• Feb 9th 2011, 10:16 AM
theodds
It is equivalent to using paramaterizing using lambda = 1 / beta in the paremeterization given on wikipedia.