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I am interested to find out the mean, median, Variance, Skewness, Kurtosis and Quantiles of the following discrete distribution with pmf (E^((1 - E^[Nu]) n) [Nu]^x BellB[n, x])/x! with x=0,1,2...... please guide me how I compute these measures using methematica? share mathematica code. Reg; Amin

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    $\begingroup$ dist = ProbabilityDistribution[(E^((1 - E^Nu) n) Nu^x BellB[n, x])/x!, {x, 0, Infinity, 1}]; then Mean[dist], Variance[dist] ... etc. However these won't work unless you provide a value for Nu and n. If Nu=1 and n=1, then Sum[(E^((1 - E^Nu) n) Nu^x BellB[n, x])/x!, {x, 0, \[Infinity]}] is E^(2 - E), so there's some normalization in n and Nu required. $\endgroup$ – flinty Jul 18 at 17:32
  • $\begingroup$ You can get a symbolic result by setting a positive integer value for n and use dist = ProbabilityDistribution[(E^((1 - E^Nu) n) Nu^x BellB[n, x])/x! /. n -> 5, {x, 0, Infinity, 1}, Method -> "Normalize"]; Mean[dist]//FullSimplify (as an example for n = 5. By trying different values of n you might see a general formula for any positive integer value of n. $\endgroup$ – JimB Jul 18 at 21:49
  • $\begingroup$ Dear JimB thanks for response. Actually I want to compute these measures in symbolic form i.e. with n and Nu $\endgroup$ – Muhammad Amin Jul 19 at 5:43
  • $\begingroup$ What makes you think there is a compact symbolic form for even the mean? $\endgroup$ – JimB Jul 19 at 6:04
  • $\begingroup$ Yes in compact symbolic form especially for Mean, Variance, and Quantile $\endgroup$ – Muhammad Amin Jul 19 at 6:10
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$Version

(* "12.1.1 for Mac OS X x86 (64-bit) (June 19, 2020)" *)

Clear["Global`*"]

dist = ProbabilityDistribution[(E^((1 - E^Nu) n) Nu^x BellB[n, x])/x!, {x, 0, 
    Infinity, 1},
   Assumptions -> {Element[n, Integers], n >= 0},
   Method -> "Normalize"];

Verifying that the distribution is valid for some typical values of n

And @@ Table[
  Sum[PDF[dist, x], {x, 0, Infinity}] == 1, {n, 0, 5}]

(* True *)

For a specific value of n

n = 3;

Join[
  {{Style[StringForm["n = ``", n], 14, Bold], SpanFromLeft}},
  {#, #[dist]} & /@ {Mean, Variance, Skewness, Kurtosis}] //
 Grid[#, Frame -> All] &

enter image description here

The median and other quartiles cannot be found symbolically.

| improve this answer | |
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  • $\begingroup$ Dear Bob thanks for your response $\endgroup$ – Muhammad Amin Sep 2 at 18:08

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