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Re: wrong solution for double integral of piecewise
*To*: mathgroup at smc.vnet.net
*Subject*: [mg97000] Re: [mg96968] wrong solution for double integral of piecewise
*From*: Bob Hanlon <hanlonr at cox.net>
*Date*: Sun, 1 Mar 2009 04:58:05 -0500 (EST)
*Reply-to*: hanlonr at cox.net
When you integrated by hand you assumed that b > a, Mathematica gives a more complicated result since it does not make that assumption unless you tell it to do so.
Clear[f]
f[x_, p_] =
Piecewise[{{k, {a <= x <= b, a <= p <= b}}}];
int = Integrate[f[x, p],
{x, -Infinity, Infinity},
{p, -Infinity, Infinity}]
Piecewise[{{k*(a - b)^2,
a - b < 0}}, 0]
Simplify[Reduce[int == 1], b > a] // ToRules
{k -> 1/(a - b)^2}
The UnitStep expression is equivalent to the Piecewise expression
expr = {k -> -(1/((a - b)^2 (-1 + UnitStep[a - b])))};
Simplify[expr, b > a]
{k -> 1/(a - b)^2}
You can include the assumption with the integral
Assuming[{b > a}, Integrate[f[x, p],
{x, -Infinity, Infinity},
{p, -Infinity, Infinity}]]
k (a-b)^2
Integrate[f[x, p],
{x, -Infinity, Infinity},
{p, -Infinity, Infinity},
Assumptions -> {b > a}]
k (a-b)^2
Bob Hanlon
---- Tom Roche <tlroche at gmail.com> wrote:
=============
wrong solution for double integral of piecewise function
I've got a function
f[\[Chi]_, \[Psi]_] =
{Piecewise[{{k, {a <= \[Chi] <= b, a <= \[Psi] <= b}}}, 0]}
I'm attempting to solve for k such that f becomes a probability
density function by applying the normalization constraint, i.e.
solving for k such that the double (indefinite) integral of f equals
1. I can do this by hand pretty easily, integrating first/inside WRT
psi and second/outside WRT chi, and I get
(1) k = 1/((b-a)^2)
I don't have the world's greatest calculus chops, but that looks
correct to me. (Am I missing something?) However, when I use
Mathematica to
Solve[
First[
Integrate[
Integrate[
f[\[Chi], \[Psi]], {\[Psi], -\[Infinity], \[Infinity]}
],
{\[Chi], -\[Infinity], \[Infinity]}
]
] == 1, k
]
I get
(2) {{k -> -(1/((a - b)^2 (-1 + UnitStep[a - b])))}}
which seems wrong to me, though I'll admit I don't know what
"UnitStep" means. So I'm wondering
* does (1) = (2)? or
* have I made a syntax error? or
* is this just really hard to solve symbolically? If so, is there a
better way to setup this function and its solution?
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