UnitStep function leads to very difficult Integration
- To: mathgroup at smc.vnet.net
- Subject: [mg48098] UnitStep function leads to very difficult Integration
- From: Nathan Moore <nmoore at physics.umn.edu>
- Date: Thu, 13 May 2004 00:08:25 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Hello all,
I'm trying to evaluate expectation values from joint probability
distributions which I've had to define in a piecewise manner, ie
norm = Integrate[G[1,x]G[2,x],{x,0,10}];
<x^2> = Integrate[G[1,x]G[2,x] x^2,{x,0,10}] * norm
where the G[k,x] is a set of polynomials defined piecewise on
intervals, {(0,2),(2,4),(4,6),etc}.
I assume the most Mathematica-friendly way to define these functions is
with the UnitStep[] function. This works to varying degree. I was
able to check the probability normalization this way (integrating only
one G[k,x]) with an exact result. Unfortunately though Mathematica
seems unable to find an exact experssion for the expectation when the
G[k,x] polynomials take sufficiently complex form.
The integration output starts to look like,
Integrate[ d^4 ( (d-3)UnitStep[3-d] UnitStep[d-1]/d + UnitStep[1 - d]
UnitStep[d])^2, {d,0,3}]
Of course I can evaluate these integrations numerically, but the
expectations seem to follow a pattern of rational numbers (1/2, 5/12,
5/6 etc...) and I'd really like to know the exact value.
Is there a more intelligent way to compose these functions? The
Simplify[] function doesn't seem to work on UnitStep[]
best regards,
Nathan Moore
University of Minnesota Physics