Statistics Questions: Weibull and Beta Distributions?

Question

These aren't my homework questions, they're what I need to know to do my homework questions:

Given a Weibull distribution with parameters a and Lambda, how do you find the median and percentiles?

Given n trials of that same distribution, how do you find expectation and variance of a certain X value (like, the number of trials in the n that are above X)?

Given a beta distribution with parameters a and b, how do you find the state space of the distribution, as well as overall expectation and variance and P(X<=Y) for some value X and Y?

In my previous stats class we only did normal distributions, so I'm a bit lost when it comes to these Weibull and Beta distributions.

Answer 1

To find the median you integrate the probability function up to the point that the value of the integral is equal to 0.5.

Similarly you do the same for the percentiles, maybe Excel has the function and will do that for you.
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Rewording your question:

Given n trials of that same distribution and a fixed number z, how many values in the n trials will on average be above z

Just integrate from o to that value z, to find the probability that of the number being below z, say this probability value is p, then
what you have is a good old friendly binomial distribution, with 1-p as the probability of a success (trial gave a value above z), and p as the probability of a value below z. The variance of a binomial experiment is np(1-p) and the expectation is n(1-p), warning success in this case has probability 1-p, so be careful plugging ing formulas.

I do not know what a state space is.

You will have to do the usual integration or look up in a statistics reference what the mean and variance of a beta distribution is, its out there.

do the integration to form the cumulative distribution function for beta.

I am rusty on this, I hope not to have made it worse.

Answer 2

I have done quite a bit of stats but they were not distributions I was immediately familiar with, though I have come acorss the Beta before. However, I did find these explanations for you:

http://en.wikipedia.org/wiki/Weibull_distribution

http://en.wikipedia.org/wiki/Beta_distribution

For the Weibull I think they use k instead of your a as the shape parameter and for the Beta, alpha and beta instead of your a and b parameters.

There are some good external links on those pages, if what you need isn't there :)

Answer 3

Here is an excellent source for different distributions.

http://www.soa.org/ccm/cms-service/stream/asset?asset_id=10959090&g11n

from the SOA website and from the book Loss Models. This is what you are provided with for the 3rd and 4th (I think?) actuarial exams.

Edit: It's a PDF. Scroll down.

Even though I took these exams years ago, I still keep a copy of this around for reference.