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From: Paulo Andrade on 18 May 2010 07:50
I am looking for a continuous probability density function defined in the interval [0, 1] with the following properties:

1. If the mode is 0.5 then the mean is also 0.5
2. If the mode is < 0.5 then: mean < mode
3. If the mode is > 0.5 then: mean > mode
4. The first derivative of the probability density function must be a continuous function.

Someone suggested me to try the Beta Probability Density Function. However, I fail to see how the Beta PDF can comply with the requirements I have. Note that the requirements are such that the mean is always in the shorter tail, and on Beta PDF the mean is, as far I know, always in the longer tail.

I also wish to have a shape parameter to control the position of the mean relative to the mode.

Can anyone suggest a PDF that would fulfill my requirements?
From: Image Analyst on 19 May 2010 13:44
"Paulo Andrade" <pandre45(a)gmail.com> wrote in message <hstutf$b4v$1(a)fred.mathworks.com>...
> I am looking for a continuous probability density function defined in the interval [0, 1] with the following properties:
>
> 1. If the mode is 0.5 then the mean is also 0.5
> 2. If the mode is < 0.5 then: mean < mode
> 3. If the mode is > 0.5 then: mean > mode
> 4. The first derivative of the probability density function must be a continuous function.
>
> Someone suggested me to try the Beta Probability Density Function. However, I fail to see how the Beta PDF can comply with the requirements I have. Note that the requirements are such that the mean is always in the shorter tail, and on Beta PDF the mean is, as far I know, always in the longer tail.
>
> I also wish to have a shape parameter to control the position of the mean relative to the mode.
>
> Can anyone suggest a PDF that would fulfill my requirements?
-----------------------------------------------------------------------------------------------------------

Paulo Andrade:
I can't see how it's possible that the mean of a distribution would be in the shorter tail.

Can you give ANY distribution where that's true? Just a simple list of counts in a vector that I can throw into bar() would be great.
Thanks,
ImageAnalyst
From: Peter Perkins on 19 May 2010 14:20
On 5/19/2010 1:44 PM, Image Analyst wrote:

> I can't see how it's possible that the mean of a distribution would be
> in the shorter tail.

You can cook up things like this:

>> x = [1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 5 6 7 8 9 10]/11
x =
Columns 1 through 8
0.090909 0.090909 0.090909 0.090909 0.090909
0.18182 0.18182 0.18182
Columns 9 through 16
0.18182 0.18182 0.27273 0.27273 0.27273
0.27273 0.27273 0.36364
Columns 17 through 24
0.36364 0.36364 0.36364 0.36364 0.36364
0.45455 0.54545 0.63636
Columns 25 through 27
0.72727 0.81818 0.90909
>> mean(x)
ans =
0.33333
>> mode(x)
ans =
0.36364

but it seems rather artificial, and not at all what the beta
distribution behaves like.
From: Paulo Andrade on 20 May 2010 13:38
Thanks for your interest. First of all, I know that the Beta PDF does not fulfill my needs. I am not looking for a solution involving Beta PDF.

With the clarification above, let me explain what I mean for "tail". For me the Mode divides the PDF into two tails (commonly one shorter than the other). Now if you think of an asymmetrical PDF there is nothing that prevents, geometrically speaking, that the area under the shorter tail be larger than than the area under the longer tail.

The triangular distribution is one that is very much used in Monte Carlo Simulation. Recently, the double triangular distribution started to be used for risk analysis (see for example: www.aacei.org/technical/rps/41R-08.pdf). Now with a double triangular distribution one can make the area under each triangle such that the the area of the taller triangle be greater that the area of the other. As such the mean of a double triangular distribution can fall inside the taller triangle and therefore inside of the sorter tail.

One of the problems with double triangular distributions is the discontinuity it presents. Therefore my request.

Hope to have clarified your question.

"Image Analyst" <imageanalyst(a)mailinator.com> wrote in message <ht1814$28e$1(a)fred.mathworks.com>...
> "Paulo Andrade" <pandre45(a)gmail.com> wrote in message <hstutf$b4v$1(a)fred.mathworks.com>...
> > I am looking for a continuous probability density function defined in the interval [0, 1] with the following properties:
> >
> > 1. If the mode is 0.5 then the mean is also 0.5
> > 2. If the mode is < 0.5 then: mean < mode
> > 3. If the mode is > 0.5 then: mean > mode
> > 4. The first derivative of the probability density function must be a continuous function.
> >
> > Someone suggested me to try the Beta Probability Density Function. However, I fail to see how the Beta PDF can comply with the requirements I have. Note that the requirements are such that the mean is always in the shorter tail, and on Beta PDF the mean is, as far I know, always in the longer tail.
> >
> > I also wish to have a shape parameter to control the position of the mean relative to the mode.
> >
> > Can anyone suggest a PDF that would fulfill my requirements?
> -----------------------------------------------------------------------------------------------------------
>
> Paulo Andrade:
> I can't see how it's possible that the mean of a distribution would be in the shorter tail.
>
> Can you give ANY distribution where that's true? Just a simple list of counts in a vector that I can throw into bar() would be great.
> Thanks,
> ImageAnalyst
From: Paulo Andrade on 26 May 2010 09:29
A simpler PDF would be one in where Mode = Mean for the interval (0, 1). Notice that 0 and 1 are not in the interval. Also the 1st derivative of the PDF must be continuous.

Can anyone help?

Peter Perkins <Peter.Perkins(a)MathRemoveThisWorks.com> wrote in message <ht1a5k$ooj$1(a)fred.mathworks.com>...
> On 5/19/2010 1:44 PM, Image Analyst wrote:
>
> > I can't see how it's possible that the mean of a distribution would be
> > in the shorter tail.
>
> You can cook up things like this:
>
> >> x = [1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 5 6 7 8 9 10]/11
> x =
> Columns 1 through 8
> 0.090909 0.090909 0.090909 0.090909 0.090909
> 0.18182 0.18182 0.18182
> Columns 9 through 16
> 0.18182 0.18182 0.27273 0.27273 0.27273
> 0.27273 0.27273 0.36364
> Columns 17 through 24
> 0.36364 0.36364 0.36364 0.36364 0.36364
> 0.45455 0.54545 0.63636
> Columns 25 through 27
> 0.72727 0.81818 0.90909
> >> mean(x)
> ans =
> 0.33333
> >> mode(x)
> ans =
> 0.36364
>
> but it seems rather artificial, and not at all what the beta
> distribution behaves like.
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