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Re: compelling evaluation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg93205] Re: [mg93175] compelling evaluation
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Fri, 31 Oct 2008 03:05:17 -0500 (EST)
  • References: <200810300701.CAA00563@smc.vnet.net>

randolph.silvers at deakin.edu.au wrote:
> I have created a function from a PDF and so it is non-zero on the unit
> interval and 0 elsewhere. But when I try to integrate some function of
> that function, it is not evaluated. How can I get it to be evaluated?
> 
> For example,
> 
> project=UniformDistribution[{0,1}];
> f[y_]:=PDF[project,y]/(CDF[project,1] - CDF[project,0]);
> F[y_]:=Integral_0^y f[z]dz;
> 
> Now, I define
> 
> pi0[y_]:= F[y]
> pi1[y_]:= Integral_y^1 (1-q) f[q] dq
> pi2[y_]:= Integral_y^1 q f[q] dq
> T[y_]:= Integral_y^1 (q-y) f[q] dq
> 
> When I enter a numerical value, each is correctly computed; when I
> differentiate, it also looks correct. For example,
> 
> D[T[y],y] returns
> 
> Integral_y^1 -{1 0<=q<=1 dq
> 
> and, T[pistar] returns
> 
> Integral_pistar^1 (-pistar + q)({1 0 <=q <= 1) dq
> 
> How can I compel Mathematica to "know" or evaluate T[pistar] and
> return the symbolic expression assuming that q is in the relevant
> domain?
> 
> Then, D[T[y],y] would return -(1-y) and T[pistar] would return:
> 
> 1/2 - pistar + pistar^2/2

You appear to be using TeX notation in places where you probably want 
Mathematica to be used instead.

I start with

project = UniformDistribution[{0,1}];
f[y_] := PDF[project,y]/(CDF[project,1] - CDF[project,0]);
T[y_] := Integrate[(q-y)*f[q], {q,y,1}]

Then one can use Assuming[...Refine[...]] to enforce the domain 
restrictions.

In[9]:= InputForm[Assuming[0<y<1, Refine[D[T[y],y]]]]
Out[9]//InputForm= (-2 + 2*y)/2

In[12]:= InputForm[Assuming[0<pistar<1, Refine[T[pistar]]]]
Out[12]//InputForm= (1 - 2*pistar + pistar^2)/2

I will note that without the domain restrictions, you can still get 
reasonable results expressed via Piecewise.

Daniel Lichtblau
Wolfram Research




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