# Modeling membership function given some survey data or empirical distribution

For example, I have a set of numbers (say 0 to 10) that are presented to 100 subjects. Each subject is asked whether the number is a small or a large number.

The results are that 100 people think zero is a small number, 70 people think one is a small number, etc.

Now I use a certain distribution, say, exponential to estimate the parameter of the distribution based on these data.

I assume people give subjective opinions and that most of them think 0 is a small number so it may be assumed that the degree of truth of zero being a small number is 1.

Finally, by multiplying the exponential probability density function (PDF) by a constant, I can scale the (PDF) such that the peak of it hits one and making it a membership function.

Does this sound right?

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Do you picked the exponential distribution because its pdf decreases smoothly from 0 to infinity ? If it fits (i.e. the error is small enough) it feel that your approach is reasonable, although I do not know any standard recipes on converting probabilistic functions to fuzzy functions. –  steffen May 3 '11 at 9:25
well, it can be any distribution. My doubt is that, since fuzzy set theory is based on possiblity, I wonder if it is appropriate to use a PDF to model a membership function. –  ltleung May 3 '11 at 12:00
The question "it is appropriate to use a PDF to model a membership function" is awesome ! I suggest to generalize the question (i.e. reformulate it) with your case as example. –  steffen May 3 '11 at 12:38