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Subject: Re: Mean of general Beta distribution
Date: Mon, 05 Jul 2010 07:06:36 -0500
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Wayne King wrote:
> "Ulrik Nash" <uwn@sam.sdu.dk> wrote in message 
> <i0seie$4a3$1@fred.mathworks.com>...
>> Hi Everyone,
>>
>> This is I suppose is more a general maths question.
>> I am working on a simulation where I would like to specify the upper 
>> and lower bounds of the Beta distribution, and at the same time be 
>> able to directly set the mean of the distribution, instead of 
>> indirectly via the shape parameters. I am aware of the mean of the 
>> Beta distribution, but only for lower and upper limits of 0 and 1. 
>> What is the general equation for the mean, involving the two shape 
>> parameters and the upper and lower limits of the distribution?
>>
>> Best
>>
>> Ulrik.
> 
> Hi Ulrik, the beta distribution is only defined on the interval (0,1). 
> The mean of the beta distribution is alpha/(alpha+beta).  I'm not quite 
> sure what you're asking in terms of lower and upper limits. The mean 
> depends directly on the two parameters and the shape of the resulting 
> pdf can vary greatly depending on the values you use for alpha and beta.

The beta distribution can be generalized to cover the interval (u0,u1) 
by transformation of variable of [(x-u0)/(u1-u0)] for x.   See Hahn & 
Shapiro, Statistical Models in Engineering, Wiley.

I don't have a closed form solution for the expected value and so on 
otomh, though; whether H&S have the generalized form in summary tables 
on continuous distributions I don't recall; it's not on the shelf here 
but would have to go find it :).

--