Mean of general Beta distribution
in [Matlab]
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From: Ulrik Nash on 5 Jul 2010 07:10 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.
From: Wayne King on 5 Jul 2010 07:45 "Ulrik Nash" <uwn(a)sam.sdu.dk> wrote in message <i0seie$4a3$1(a)fred.mathworks.com>... > Hi Everyone, > > This is I suppose is more a general maths question. > > I am working on a simulation where I would like to specify the upper and lower bounds of the Beta distribution, and at the same time be able to directly set the mean of the distribution, instead of indirectly via the shape parameters. I am aware of the mean of the Beta distribution, but only for lower and upper limits of 0 and 1. What is the general equation for the mean, involving the two shape parameters and the upper and lower limits of the distribution? > > Best > > Ulrik. Hi Ulrik, the beta distribution is only defined on the interval (0,1). The mean of the beta distribution is alpha/(alpha+beta). I'm not quite sure what you're asking in terms of lower and upper limits. The mean depends directly on the two parameters and the shape of the resulting pdf can vary greatly depending on the values you use for alpha and beta. Wayne
From: Ulrik Nash on 5 Jul 2010 08:09 "Wayne King" <wmkingty(a)gmail.com> wrote in message <i0sgk2$gc5$1(a)fred.mathworks.com>... > "Ulrik Nash" <uwn(a)sam.sdu.dk> wrote in message <i0seie$4a3$1(a)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.
From: dpb on 5 Jul 2010 08:06 Wayne King wrote: > "Ulrik Nash" <uwn(a)sam.sdu.dk> wrote in message > <i0seie$4a3$1(a)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 :). --
From: John D'Errico on 5 Jul 2010 08:14
"Ulrik Nash" <uwn(a)sam.sdu.dk> wrote in message <i0seie$4af$1(a)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? > This is something you CAN figure out yourself, rather trivially. If you know that the mean of a given beta distribution on the interval [0,1], with shape parameters (alpha, beta) is mu, then what is the mean of the beta with those same shape parameters on the interval [a,a+1]? I'll give you a hint. A mean is an expectation, i..e, an integral. So everything is nice and linear. (Second hint: add a.) Next, how does changing the width of the interval affect the mean? (Hint, multiply by the interval width. Again, its just an integral.) These are things that should be obvious to you. If they are not so, and if you are doing any work with simulations, you will greatly benefit from some study. John |