Re: RandomArray from user defined distribution?

*To*: mathgroup at smc.vnet.net*Subject*: [mg73390] Re: RandomArray from user defined distribution?*From*: Roland Franzius <roland.franzius at uos.de>*Date*: Thu, 15 Feb 2007 04:59:32 -0500 (EST)*Organization*: Universitaet Hannover*References*: <eqpdv5$eip$1@smc.vnet.net> <equndr$idu$1@smc.vnet.net>

roland franzius wrote: >> rob wrote: >>> I'd like to use the RandomArray to produce some data from >>> what I think is a Poisson distribution in t >>> P[t] = a^2 t Exp[-a*t] where a is mean, sigma. >>> >>> I see one can use RandomArray to produce sample data from a >>> lot of continuous distributions but the Poisson isn't among >>> them (it's only available in the discrete form). >>> >>> I've made a bunch of crippled attempts to force my P[t] to >>> put out examples but have failed. Any suggestions? Thanks. >>> >> >> Your density is a GammaDistribution[2,3], se e.g. >> >> http://mathworld.wolfram.com/GammaDistribution.html >Sir, I email you directly because I can't seem to get my posts accepted >every time on mathgroup. > >Thank you for the suggestion on the Gamma function which is definitely >what I was looking for and had not found in my searches. > >It appears I need GammaDistribution[2, mean/2] to get what I want. > >On your other suggestions, I don't follow. Please, what do you mean by >? >choose:=ip[Random[Real,{0,1}]] Hi, (*Perhaps this short example will enable you to apply the method*) (*an exotic distibution density*) density[x_] = (#/Integrate[#, {x, 0, ?}] &)[Sin[x^2]^2*Exp[-x^2/12]] Plot[density[x], {x, 0, 12}, PlotRange -> All, PlotLabel -> "density"]; (*its cumulative distribution function not invertible by algebraic methods *) distribution[x_] = Integrate[density[y], {y, 0, x}] (*the x -> F(x) plot*) Plot[distribution[x], {x,0, 8}, PlotRange -> {0, 1}, PlotLabel -> "distibution"] (* a 1024 point table of points of its inverse y -> F^(-1)(y), chopping removes small imaginary parts, the small additional increment table separates points in the first argument*) table = Chop@N@Table[ {distribution[x], x}, {x, 0, 12, 12/1024}] + Table[{x*10^-7, 0}, {x, 0, 1, 1/1024}]; (* ListPlot showing the inverted graph*) ListPlot[table, PlotJoined -> True, PlotLabel -> "inverse table", PlotRange -> {0, 10}]; (*construct an interpolating pure function Ip from that table*) Ip = Interpolation[table] (*shows the same smooth graph *) Plot[Ip[x], {x, 0.0, 1.0}, PlotLabel -> "inverse interpolation", PlotRange -> {0, 10}]; (*define a random variable from the uniform distribution on (0, 1) using \ Random[]*) choose := Ip[Random[]] sample = Table[choose, {10000}] sorted = Sort[sample]; (* ListPlots sorted and unsorted show a smooth approximation of the distribution and the described randomness of raw sample data*) sorted = Sort[Table[choose, {10000}]]; ListPlot[sorted, PlotLabel -> "sorted sample", PlotRange -> {0, 10}]; ListPlot[sample, PlotLabel -> "unsorted sample", PlotRange -> {0, 10}] Hope it helps -- Roland Franzius