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Re: Inconsistent behaviour of StudentTDistribution

  • To: mathgroup at smc.vnet.net
  • Subject: [mg103186] Re: Inconsistent behaviour of StudentTDistribution
  • From: "Sjoerd C. de Vries" <sjoerd.c.devries at gmail.com>
  • Date: Thu, 10 Sep 2009 07:19:14 -0400 (EDT)
  • References: <h87pi3$5hp$1@smc.vnet.net>

I can imagine the 2nd one is a bit confusing bit it looks like the two
are the same:

In[188]:= Assuming[x > 0 && f > 0 && f \[Element] Integers,
 D[CDF[StudentTDistribution[f], x], x] // FullSimplify]

Out[188]= (f^(f/2) (f + x^2)^(-(1/2) - f/2))/Beta[f/2, 1/2] +
 1/2 BetaRegularized[f/(f + x^2), 1, f/2, 1/2]
\!\(\*SuperscriptBox["Sign", "\[Prime]",
MultilineFunction->None]\)[x]

The second part contains term that is a derivative of a Sign function
which is zero everywhere except for zero where I'd say it's undefined.
So if we throw out the second part (something Mathematica doesn't seem
to dare) we're left with the first part, and this part is equal to the
one you get with the PDF (which you have to write as PDF
[StudentTDistribution[f], x][x], otherwise it would be a pure
function).

You can check their equivalence by plotting the functions.

I agree with you that it'd be better if Mathematica would handle this
case more elegantly.

Cheers -- Sjoerd


On Sep 9, 10:38 am, Alexey <lehi... at gmail.com> wrote:
> Hello,
> Consider the following:
>
> PDF[StudentTDistribution[f], x]
> D[CDF[StudentTDistribution[f], x], x]
>
> The outputs of these expressions must be equal. But in really the
> second is useless and confusing.



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