I am looking for a continuous probability density function defined in the interval [0, 1] with the following properties:
- If the mode is 0.5 then the mean is also 0.5
- If the mode is < 0.5 then: mean < mode
- If the mode is > 0.5 then: mean > mode
- 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?
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.
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.
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.