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Could anyone come up with a probability density function which is:

  • supported on [1,∞) (or [0,∞))
  • increasing
  • discrete
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you mean the density function? CDF is always increasing/non-decreasing. – Seyhmus Güngören Mar 17 '13 at 15:47
@Seyhmus, yes . – Yariv Mar 17 '13 at 15:48
One cannot find such a density function: The integral cannot be $1$. – André Nicolas Mar 17 '13 at 16:07
up vote 1 down vote accepted

There exist no such densities. Because, if a function is increasing, then either it has a limit and the function converges to this limit, or it doesnt have a limit.

If it doesnt have a limit then without any doubt the area under the function can not add up to $1$.

In case it is a convergent function and say it converges to $\alpha$, then an amount of area, say $\beta$ is repeated infinitely many times in the integration, therefore integral does not converge.

As a result, no such densities exist.

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@Yariv Yes. Think about the uniform (discrete or continuous) distribution. It even does not increase but does not exists either. – Seyhmus Güngören Mar 17 '13 at 16:18

A Dirac delta function at infinity.

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I was having the same question, but in a bounded environment, so the distribution would be increasing with support a and b. It grows like a slow exponential, actually is the the distribution of the following:

Take N repetitions of 4 uniformly generated values between A and B. Make the histogram of the max in each repetition. You have something increasing that is not linear, it would be a kind of low speed growing exponential.

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