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Re: Solve for parameters of a

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122957] Re: Solve for parameters of a
  • From: Barrie Stokes <Barrie.Stokes at newcastle.edu.au>
  • Date: Fri, 18 Nov 2011 06:23:58 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <201111151050.FAA23783@smc.vnet.net>

I still like my animated Plot3D with the Opacity[0.5] plane. Go on, put that with your fancy matching coloured red variance and matching blue mean and your ViewPoint report, and you can have the pot!

Barrie

>>> On 18/11/2011 at 11:21 am, in message <op.v438ldxutgfoz2 at bobbys-imac.local>,
DrMajorBob <btreat1 at austin.rr.com> wrote:
> I'll meet that and raise you this:
> 
> X = TruncatedDistribution[{0, \[Infinity]},
>     NormalDistribution[\[Mu], \[Sigma]]];
> {mean, var} = FullSimplify@Through[{Mean, Variance}@X];
> 
> Manipulate[p = {3.3532, .2242, .3637};
>   Column@{Row@{Style["variance = ", FontFamily -> "Arial",
>        FontSize -> 11.5, Red],
>       Style[ToString@height, FontFamily -> "Arial", FontSize -> 11.5],
>       Style[" = mean ", FontFamily -> "Arial", FontSize -> 11.5,
>        Blue]}, Show[{ContourPlot[
>        mean == height, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>        ContourStyle -> {Red}, ImageSize -> 350],
>       ContourPlot[var == height, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>         ContourStyle -> {Blue}]}, FrameLabel -> {"\[Mu]", "\[Sigma]"}],
>      Plot3D[{mean, var, 1}, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>      PlotStyle -> {Blue, Green, {Opacity[0.5], Gray}},
>      ImageSize -> 350, ViewPoint -> Dynamic[p]],
>     ViewPoint -> Dynamic[p]}, {{height, 1}, 0.1, 3, 0.001}]
> 
> Bobby
> 
> On Thu, 17 Nov 2011 17:23:10 -0600, Barrie Stokes  
> <Barrie.Stokes at newcastle.edu.au> wrote:
> 
>> Hi Bobby
>>
>> Very nice.  My final mod of your mod of ...
>>
>> Manipulate[
>>    Column@{Style[
>>          "mean equals variance equals " <>
>>            ToString[height], FontFamily -> "Arial", FontSize -> 11.5],
>>        Show[{ContourPlot[
>>              mean == height, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>>              ContourStyle -> {Red}, ImageSize -> 350],
>>     ContourPlot[var == height, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>>               ContourStyle -> {Blue}]},
>>     FrameLabel -> {"\[Mu]", "\[Sigma]"}],
>>         Plot3D[{mean, var, height}, {\[Mu], 0.01, 3}, {\[Sigma],
>>      0.01, 3},
>>          PlotStyle -> {Blue, Green, {Opacity[0.5], Gray}},
>>     ImageSize -> 350]}, {{height, 1},
>>      0.1, 3, 0.001}]
>>
>>
>> Best,
>>
>> Barrie
>>
>>>>> On 18/11/2011 at 6:19 am, in message  
>>>>> <op.v43um9zhtgfoz2 at bobbys-imac.local>,
>> DrMajorBob <btreat1 at austin.rr.com> wrote:
>>> Try this:
>>>
>>> X = TruncatedDistribution[{0, \[Infinity]},
>>>     NormalDistribution[\[Mu], \[Sigma]]];
>>> {mean, var} = FullSimplify@Through[{Mean, Variance}@X];
>>>
>>> Manipulate[
>>>   p = {3.3532, .2242, .3637};
>>>   Column@{Style[
>>>      "mean can equal variance if contours intersect at height = " <>
>>>       ToString[height], FontFamily -> "Arial", FontSize -> 11.5],
>>>     Show[{ContourPlot[
>>>        mean == height, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>>>        ContourStyle -> {Red}, ImageSize -> 350],
>>>       ContourPlot[var == height, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>>>         ContourStyle -> {Blue}]}, FrameLabel -> {"\[Mu]", "\[Sigma]"}],
>>>      Plot3D[{mean, var, 1}, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>>>      PlotStyle -> {Blue, Green, Gray}, ImageSize -> 350,
>>>      ViewPoint -> Dynamic[p]], ViewPoint -> Dynamic[p]}, {{height, 1},
>>>    0.1, 3, 0.001}]
>>>
>>> I think the contour at 2 only adds confusion.
>>>
>>> Bobby
>>>
>>> On Thu, 17 Nov 2011 05:03:57 -0600, Barrie Stokes
>>> <Barrie.Stokes at newcastle.edu.au> wrote:
>>>
>>>> Andrzej , Bobby
>>>>
>>>> Speaking of nice graphics:
>>>>
>>>> If you combine Bobby's mods to my ContourPlot ...
>>>>
>>>> X = TruncatedDistribution[{0, \[Infinity]},
>>>>        NormalDistribution[\[Mu], \[Sigma]]];
>>>> {mean, var} = FullSimplify@Through[{Mean, Variance}@X];
>>>> Manipulate[
>>>>  Column@{Style[ "contour height is " <> ToString[ height],
>>>>     FontFamily -> "Arial", FontSize -> 11.5 ],
>>>>        Show[{ContourPlot[
>>>>              mean == height, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>>>>              ContourStyle -> {Red}, ImageSize -> 350],
>>>>     ContourPlot[var == height, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>>>>               ContourStyle -> {Blue}]},
>>>>          FrameLabel -> {"\[Mu]", "\[Sigma]"}]}, {{height, 1}, 0.1, 3,
>>>>      0.001}]
>>>>
>>>> with  these plots from Andrzej's code (rotate both plots to reveal the
>>>> "underside") ...
>>>>
>>>> X = TruncatedDistribution[{0, \[Infinity]},
>>>>       NormalDistribution[\[Mu], \[Sigma]]];
>>>> m = Mean[X];
>>>> v = Variance[X];
>>>>
>>>> and then ...
>>>>
>>>> Plot3D[{m, v, 1}, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>>>>   PlotStyle -> {Blue, Green, Gray}]
>>>>
>>>> and ...
>>>>
>>>> Plot3D[{m, v, 2}, {\[Mu], 0.01, 3}, {\[Sigma], 0.01, 3},
>>>>   PlotStyle -> {Blue, Green, Gray}]
>>>>
>>>> the difference between height=1 and height=2 is clearly revealed.
>>>>
>>>> Cheers
>>>>
>>>> Barrie
>>>>
>>>>
>>>>
>>>>
>>>>>>> On 16/11/2011 at 8:46 pm, in message
>>>>>>> <201111160946.EAA06190 at smc.vnet.net>,
>>>> Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
>>>>
>>>>> On 15 Nov 2011, at 11:50, paul wrote:
>>>>>
>>>>>> I'm trying to solve the following problem:
>>>>>> X = TruncatedDistribution[{0, \[Infinity]},
>>>>>>  NormalDistribution[\[Mu], \[Sigma]]]
>>>>>> Solve[Mean[X] == 1 && Variance[X] == 1, {\[Mu], \[Sigma]}, Reals]
>>>>>>
>>>>>> I get an error message: "This system cannot be solved with the  
>>>>>> methods
>>>>>> available to Solve." It doesn't help if I replace Solve with NSolve.
>>>>>>
>>>>>> In case I've made a mistake in defining the problem, I should say  
>>>>>> that
>>>>>> I'm looking for the parameters of a normal distribution so that, if
>>>>>> the normal is truncated on the left at zero, the result will be a
>>>>>> truncated distribution whose mean and variance are both 1. It seems  
>>>>>> to
>>>>>> me Mathematica should be able to solve this, at least numerically.
>>>>>>
>>>>>> Many thanks for any suggestions.
>>>>>>
>>>>>>
>>>>>
>>>>> Your first mistake is to use functions (Solve and NSolve) which are  
>>>>> not
>>>>> intended for such purposes at all. NSolve can only solve (numerically)
>>>>> polynomial equations and systems of such. Your equations are certainly
>>>>> not of
>>>>> this kind. Solve (in version 8) can also solve certain univariate
>>>>> transcendental equations but not systems of such. So again, there is  
>>>>> no
>>>>> point
>>>>> at all of trying either of these functions on your system.
>>>>>
>>>>> The only function that might work is FindRoot. However, before one  
>>>>> even
>>>>> starts, one has to have some reason for believing such a solution
>>>>> exists.
>>>>> Now, looking at the graphs below, I see no such reason. So do you have
>>>>> one?
>>>>>
>>>>> X = TruncatedDistribution[{0, \[Infinity]},
>>>>>    NormalDistribution[\[Mu], \[Sigma]]];
>>>>>
>>>>> m = Mean[X];
>>>>>
>>>>> v = Variance[X];
>>>>>
>>>>> Plot3D[{m, v, 1}, {\[Mu], 0.1, 2}, {\[Sigma], 0.1, 2},
>>>>>  PlotStyle -> {Blue, Green, Black}]
>>>>>
>>>>> Andrzej Kozlowski
>>>>
>>>>
>>>
> 



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