       TransformedDistribution

• To: mathgroup at smc.vnet.net
• Subject: [mg116874] TransformedDistribution
• From: Jagra <jagra24891 at mypacks.net>
• Date: Thu, 3 Mar 2011 05:56:52 -0500 (EST)

```I have developed some code to generate random variates from the
product of a LogNormalDistribution and a StableDistribution:

LNStableRV[{\[Alpha]_, \[Beta]_, \[Gamma]_, \[Sigma]_, \[Delta]_},

n_] := Module[{LNRV, SDRV, LNSRV},
LNRV = RandomVariate[LogNormalDistribution[Log[\[Gamma]], \[Sigma]],
n];
SDRV = RandomVariate[
StableDistribution[\[Alpha], \[Beta], \[Gamma], \[Sigma]], n];
LNRV * SDRV + \[Delta]
]

(* Note the delta serves as a location parameter *)

I think this works fine:

LNStableRV[{1.5, 1, 1, 0.5, 1}, 50000];
Histogram[%, Automatic, "ProbabilityDensity",
PlotRange -> {{-4, 6}, All}, ImageSize -> 250]
ListPlot[%%, Joined -> True, PlotRange -> All]

Now I'd like to create a TransformedDistribution along the same lines
so that I can use PDF[], CDF[], etc. on this custom distribution and
easily do plots and other analysis.

Extrapolating from an example in Documentation Center >
TransformedDistribution:

\[ScriptCapitalD] =
TransformedDistribution[
u v, {u \[Distributed] ExponentialDistribution[1/2],
v \[Distributed] ExponentialDistribution[1/3]}];

I've tried this:

LogNormalStableDistribution[\[Alpha]_, \[Beta]_, \[Gamma]_, \
\[Sigma]_, \[Delta]_] := Module[{u, v},
TransformedDistribution[
u * v + \[Delta], {u \[Distributed]
LogNormalDistribution[Log[\[Gamma]], \[Sigma]],
v \[Distributed]
StableDistribution[\[Alpha], \[Beta], \[Gamma], \[Sigma]]}]
];

\[ScriptCapitalD] = LogNormalStableDistribution[1.5, 1, 1, 0.5, 1]

Which gives me this:

TransformedDistribution[
1 + \[FormalX]1 \[FormalX]2, {\[FormalX]1 \[Distributed]
LogNormalDistribution[0, 0.5], \[FormalX]2 \[Distributed]
StableDistribution[1, 1.5, 1, 1, 0.5]}]

But when I try to plot a PDF of the distribution it never seems to
finish (granted I haven't let it run more than 10 minutes so far):

Plot[PDF[\[ScriptCapitalD], x], {x, -4, 6}] (* This should plot over
the same range as the Histogram above *)

So, some questions:

Does my function: LogNormalStableDistribution[] make sense to do this
kind of thing?

If yes do I:

Just need to let the Plot[] run longer?
Change it somehow?
What can I do to make it run faster?

If not:

Do I need to approach this in a different way?
Use MixtureDistribution?
Use Something else?
Any ideas appreciated.

Best,

J

```

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