Re: Defining a Discrete Probability Distribution using ProbabilityDistribution

*To*: mathgroup at smc.vnet.net*Subject*: [mg122531] Re: Defining a Discrete Probability Distribution using ProbabilityDistribution*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Mon, 31 Oct 2011 06:50:46 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

Look at the example under Scope>Discrete Univariate Distributions: \[ScriptCapitalD] = ProbabilityDistribution[16^(2 x + 1)/(Sinh[16] (2 x + 1)!), {x, 0, \[Infinity], 1}]; DiscretePlot[Evaluate@PDF[\[ScriptCapitalD], x], {x, 0, 15}, ExtentSize -> 1/2] Now draw the very same function with Plot: Plot[Evaluate@PDF[\[ScriptCapitalD], x], {x, 0, 15}] This, as you can see, is NOT a discrete pdf. This plot LOOKS like a discrete one: DiscretePlot[Evaluate@CDF[\[ScriptCapitalD], x], {x, 0, 15}, ExtentSize -> Right] What we need is the heights of those bars: heights = CDF[\[ScriptCapitalD], Range[0, 16]] // N BarChart[heights] {3.60113*10^-6, 0.000157249, 0.00212394, 0.0141114, 0.0567335, \ 0.155927, 0.318706, 0.517141, 0.703903, 0.843702, 0.928913, 0.972023, \ 0.990417, 0.997125, 0.999239, 0.999821, 0.999963} Weights at the integers are given by weights = Thread[{Range[0, 15], Differences@heights}] {{0, 0.000153648}, {1, 0.00196669}, {2, 0.0119875}, {3, 0.0426221}, {4, 0.0991933}, {5, 0.162779}, {6, 0.198435}, {7, 0.186762}, {8, 0.139799}, {9, 0.0852107}, {10, 0.0431105}, {11, 0.0183938}, {12, 0.00670772}, {13, 0.00211475}, {14, 0.000582125}, {15, 0.000141121}} or Clear[weight] SetAttributes[weight, Listable] weight[-1] = 0; weight[n_Integer?NonNegative] := weight[n] = CDF[\[ScriptCapitalD], n] - CDF[\[ScriptCapitalD], n - 1] ListPlot[weight@Range[0, 20]] Bobby On Sun, 30 Oct 2011 04:23:09 -0500, Jonathan Foley <jonefoley at gmail.com> wrote: > > > I'm trying to use ProbabilityDistribution (http:// > reference.wolfram.com/mathematica/ref/ProbabilityDistribution.html) to > define a discrete probability distribution given a pdf. The problem is > I can't get it to actually be discrete: > > In: twostatedist = > ProbabilityDistribution[(Gamma[ > a/d + N]/(Gamma[N + 1]*Gamma[a/d + b/d + N])*(Gamma[a/d + b/d]/ > Gamma[a/d]) (c/d)^N* > Hypergeometric1F1[a/d + N, a/d + b/d + N, -c/d]), {N, 0, > Infinity, 1}, Assumptions -> {N \[Element] Integers && N >= 0}]; > params = {a -> 0.3, b -> 60/25, c -> 60, d -> 0.2}; > PDF[twostatedist /. params, 0.5] > > Out: 0.00976739 > > > This should yield a probability of 0 but it doesn't. Based on the > documentation for a discrete distribution you are supposed to use the > form ProbabilityDistribution[pdf,{x,xmin,xmax,dx}]. It is not > explained what dx is, but I assumed this was the step size, so I set > it to 1. I further tried to enforce discreteness by using Assumptions, > but this doesn't appear to work. > > Does anyone know how to do this? > -- DrMajorBob at yahoo.com