Re: approximate a pdf function with exponential distribution

From: Robert Israel (israel_at_math.ubc.ca)
Date: 01/11/05 

Date: 11 Jan 2005 06:26:37 GMT

In article <41E34101.70657A9E@virginia.edu>,
Julian V. Noble <jvn@virginia.edu> wrote:

>Yan ZHANG wrote:

>> I have got the probability density function(pdf) of a continuous
>> non-negative random variable. Now, I am hoping to approximate this pdf
>> function with an exponential distributed random variable.
 
>> 1. in such problem, which criteria is the quantity we should minimize?

That's likely to depend on what your purpose is in making this
approximation. There are lots of plausible criteria.

>If the distribution is exponential,

> dp(x) = A exp(-Ax) dx ,

>then the mean is 1/A and the variance is 1/A^2 .

>Since this is a 1-parameter distribution you can't fit both the
>mean and variance of your pdf. However, if you use the distribution

> dp(x) = A^{\alpha + 1} exp(-Ax) dx / \Gamma( \alpha + 1)

>you have two parameters to play with and can thus fit these
>aspects of your pdf.

Huh? That's not a probability measure.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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