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Math Help - Probability Density Function needed

  1. #1
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    Question Probability Density Function needed

    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.
    Last edited by pandre; May 15th 2010 at 02:05 PM. Reason: Complementing
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by pandre View Post
    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
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  3. #3
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    Red face

    Quote Originally Posted by CaptainBlack View Post
    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
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  4. #4
    MHF Contributor matheagle's Avatar
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    The mean is {\alpha\over \alpha+\beta} what do you have as the mode?
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  5. #5
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    Quote Originally Posted by matheagle View Post
    The mean is {\alpha\over \alpha+\beta} what do you have as the mode?
    I am using the mode as
    <br />
{\alpha-1\over\alpha+\beta-2}<br />

    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
    Last edited by pandre; May 17th 2010 at 11:51 AM. Reason: Latex error
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  6. #6
    MHF Contributor matheagle's Avatar
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    Ok, I just checked that.
    I get that as the mode too.
    Last edited by matheagle; May 17th 2010 at 06:54 PM.
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  7. #7
    MHF Contributor matheagle's Avatar
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    yup the Captain's right.

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

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

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

    so, adding a to both sides we have 2a< a+b or {a\over a+b}<1/2
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  8. #8
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    Question

    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.
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by pandre View Post
    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.
     <br />
\alpha<\beta \Rightarrow mode<0.5<br />

     <br />
\alpha+\beta>2 \Rightarrow mode<mean<br />

    and a similar construction will give the other case/s

    CB
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  10. #10
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    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.
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  11. #11
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    Captain and Matheagle, once again I would like to tell that I am grateful for your efforts in helping me.

    You may be imagining why I need such a distribution, and I will explain my motivation. First of all, I concluded that the Beta PDF will 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 shorter tail.

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

    Hope to have clarified my motivation.
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  12. #12
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    Question

    A particular and much 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?
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