# Probability Density Function needed

#### pandre

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.
I also wish to be able to have a shape parameter to control the distance between the mean and the mode

It would be even better if I could control the position of the mean relative to the mode via the parameter.

Does anyone knows such a function?

Thanks.

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#### CaptainBlack

MHF Hall of Fame
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.
I also wish to be able to have a shape parameter to control the distance between the mean and the mode

It would be even better if I could control the position of the mean relative to the mode via the parameter.

Does anyone knows such a function?

Thanks.
I would look at the Beta distribution if I were you.

CB

• matheagle and pandre

#### pandre

I would look at the Beta distribution if I were you.

CB
Thanks. I have been working with Beta but I could not find a set of rules or an algorithm for the α and β parameters that would fulfill the requirements.
Could you give it a try?
Best regards
Paulo

#### matheagle

MHF Hall of Honor
The mean is $$\displaystyle {\alpha\over \alpha+\beta}$$ what do you have as the mode?

• pandre

#### pandre

The mean is $$\displaystyle {\alpha\over \alpha+\beta}$$ what do you have as the mode?
I am using the mode as
$$\displaystyle {\alpha-1\over\alpha+\beta-2}$$

I fail to see how the Beta Probability Density Function can comply to 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.

Regards,

Paulo

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#### matheagle

MHF Hall of Honor
Ok, I just checked that.
I get that as the mode too.

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#### matheagle

MHF Hall of Honor
yup the Captain's right.

If $$\displaystyle {a\over a+b}=1/2$$ then $$\displaystyle a=b$$

So $$\displaystyle {a-1\over 2a-2}=1/2$$.

And if $$\displaystyle {a-1\over a+b-2}<1/2$$ then $$\displaystyle a<b$$

so, adding a to both sides we have $$\displaystyle 2a< a+b$$ or $$\displaystyle {a\over a+b}<1/2$$

#### pandre

Matheagle,
thanks for the effort, but I knew this and it does not fulfill the requirements.
I know that, for the Beta PDF, Mode < 0.5, implies Mean < 0.5. However this also implies that Mean > Mode and what I am looking for is that, in this case, Mean < Mode. And the Beta PDF cannot, as far as I see, fulfill the requirements.

#### CaptainBlack

MHF Hall of Fame
Matheagle,
thanks for the effort, but I knew this and it does not fulfill the requirements.
I know that, for the Beta PDF, Mode < 0.5, implies Mean < 0.5. However this also implies that Mean > Mode and what I am looking for is that, in this case, Mean < Mode. And the Beta PDF cannot, as far as I see, fulfill the requirements.
$$\displaystyle \alpha<\beta \Rightarrow mode<0.5$$

$$\displaystyle \alpha+\beta>2 \Rightarrow mode<mean$$

and a similar construction will give the other case/s

CB

• pandre

#### pandre

Captain,

That is the inverse of what I am looking for!!

I require that in this case Mode > Mean

Another way of saying it is that for Mode < 0.5 I require that the Mean is at the left of the Mode.