# Any one recognize this probabilty equation?

**URL:** <https://boards.straightdope.com/t/any-one-recognize-this-probabilty-equation/434377>\
**Category:** Factual Questions\
**Created:** [January 21, 2008, 2:45pm UTC](https://boards.straightdope.com/t/any-one-recognize-this-probabilty-equation/434377 "2008-01-21T14:45:29Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![Grey](https://avatars.discourse-cdn.com/v4/letter/g/b782af/32.png) [@Grey](https://boards.straightdope.com/u/Grey)\
**Post date:** [January 21, 2008, 2:45pm UTC](https://boards.straightdope.com/t/any-one-recognize-this-probabilty-equation/434377/1 "2008-01-21T14:45:29Z")

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I’ve been passed a probability density function that I don’t recognize. Worse, I’ve not been given any values for the various constants. I’ve gone digging but can’t seem to identify it so I turn to the Dope.

p(x) = F/2 for x \<= T

p(x) = (1-F)(B-1) / X[sub]m[/sub] \* (x/X[sub]m[/sub])^(-B) for x\>T, where T=2

Any ideas?

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**Author:** ![don\_t\_ask](https://avatars.discourse-cdn.com/v4/letter/d/e68b1a/32.png) [@don\_t\_ask](https://boards.straightdope.com/u/don_t_ask)\
**Post date:** [January 21, 2008, 2:46pm UTC](https://boards.straightdope.com/t/any-one-recognize-this-probabilty-equation/434377/2 "2008-01-21T14:46:31Z")

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Just guess.

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [January 21, 2008, 3:44pm UTC](https://boards.straightdope.com/t/any-one-recognize-this-probabilty-equation/434377/3 "2008-01-21T15:44:51Z")

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As specified, the cdf is not valid because there’s no point at which it’s equal to zero. Was there an additional constrain that p(x) = 0 for x \< 0? Regardless, you’re working with a mixed distribution here, so it’s not going to be any of the distributions you’d normally see in textbooks.

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**Author:** ![Omphaloskeptic](https://avatars.discourse-cdn.com/v4/letter/o/bcef8e/32.png) [@Omphaloskeptic](https://boards.straightdope.com/u/Omphaloskeptic)\
**Post date:** [January 21, 2008, 6:31pm UTC](https://boards.straightdope.com/t/any-one-recognize-this-probabilty-equation/434377/4 "2008-01-21T18:31:46Z")

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**ultrafilter** is right about the poorly specified domain.

If this pdf is only valid for x\>0, then there are a few things you can figure out. First, the distribution is a mixture of a uniform distribution (on [0,T]) and a power-law distribution (on [T,infinity)). You specify T=2, so integrating the uniform distribution gives a probability of (F/2)\*T=F. This tells you the interpretation of F: with probability F you’re choosing from the uniform distribution, and otherwise you choose from the power law.

Now integrating the power-law distribution only gives a finite answer for B\>1; in this case it integrates to (1-F)(T/X[sub]m[/sub])[sup]1-B[/sup]. We know that it should integrate to 1-F, though, so we must have X[sub]m[/sub]=T=2.

If you did not have T=2, the requirement that the density be normalized would give a more complicated constraint on the four parameters.