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Thread Subject:
Mean of general Beta distribution

Subject: Mean of general Beta distribution

From: Ulrik Nash

Date: 5 Jul, 2010 11:10:06

Message: 1 of 6

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.

Subject: Mean of general Beta distribution

From: Wayne King

Date: 5 Jul, 2010 11:45:06

Message: 2 of 6

"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.

Wayne

Subject: Mean of general Beta distribution

From: Ulrik Nash

Date: 5 Jul, 2010 12:09:03

Message: 3 of 6

"Wayne King" <wmkingty@gmail.com> wrote in message <i0sgk2$gc5$1@fred.mathworks.com>...
> "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.
>
> Wayne

Hi Wayne,

Actually, there is a more general version of the Beta, or perhaps more commonly known as the 4-parameter Beta distribution. There is mention of this here:

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

Best regards,

Ulrik.

Subject: Mean of general Beta distribution

From: dpb

Date: 5 Jul, 2010 12:06:36

Message: 4 of 6

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 :).

--

Subject: Mean of general Beta distribution

From: John D'Errico

Date: 5 Jul, 2010 12:21:04

Message: 5 of 6

"Wayne King" <wmkingty@gmail.com> wrote in message <i0sgk2$gc5$1@fred.mathworks.com>...
> "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).

Actually, this is not true.

There are many who use a 4 parameter beta distribution,
wherein the 3rd and 4th parameters are the lower and
upper limits of the distribution support. It is a simple
shift and scale transformation from the standard beta.

And of course, a beta distribution is based on a shifted
incomplete beta function anyway, so it is already really
a transformation based on a shift and scale.

I'll bet this is discussed in the bible of distributions,
i.e., Johnson, Kotz, & Balakrishnan.

John

Subject: Mean of general Beta distribution

From: Wayne King

Date: 5 Jul, 2010 12:56:14

Message: 6 of 6

"John D'Errico" <woodchips@rochester.rr.com> wrote in message <i0sing$37i$1@fred.mathworks.com>...
> "Wayne King" <wmkingty@gmail.com> wrote in message <i0sgk2$gc5$1@fred.mathworks.com>...
> > "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).
>
> Actually, this is not true.
>
> There are many who use a 4 parameter beta distribution,
> wherein the 3rd and 4th parameters are the lower and
> upper limits of the distribution support. It is a simple
> shift and scale transformation from the standard beta.
>
> And of course, a beta distribution is based on a shifted
> incomplete beta function anyway, so it is already really
> a transformation based on a shift and scale.
>
> I'll bet this is discussed in the bible of distributions,
> i.e., Johnson, Kotz, & Balakrishnan.
>
> John

Sorry all, I stand corrected. Thanks for the info!
Wayne

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