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Specify a general rule, so that mathematica always evaluate it?

Posted 11 months ago
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I want to specify a set of some general mathematical rules regarding probability theory to solve equations with "expectation" and "variance" for random variables. Therefore I want to "teach" mathematica to evaluate the expression

Integrate[p[x], {x, -Infinity, Infinity}] -> 1

always to "1". p[x] should be any arbitrary "Probability Density Function" of "x" with the result, whenever mathematica's evaluation finds the integral in an expression, it should replace it by "1" I tried this with

Simplify[Integrate[p[x], {x, -Infinity, Infinity}] /. Integrate[p[x], {x, -Infinity, Infinity}] :> 1]

and similar expressions and rules, but when I put into the next cell:

Integrate[p[x], {x, -Infinity, Infinity}]

the output is always the Integral itself, but not the expected value of "1".

My final goal is if I specify:

expectation[x_] := Integrate[x*p[x], {x, -Infinity, Infinity}]
expectation[z_] := Integrate[z*p[x], {x, -Infinity, Infinity}] /; FreeQ[z, x] := z /; FreeQ[z, x]

that mathematica recognize "z" as a constant, independant from x, get it out of the Integral and evaluates the remaining Integral to "1" and the final result as "z"

How can I solve such a problem ? Thank you for your support.

Regards Uwe

2 Replies

Uwe,

you can specify an upvalue for p:

p /: Integrate[p[x_], {x_, -Infinity, Infinity}] := 1

Now the only problem is that Integrate does not factor constants out of integrals so your rule will not trigger in the case of z*p[x] in the integral. To make this happen you can define a function to extract constants from integrals (from this stackexchange post)

moveconst[
  x_] := (x /. 
   Integrate[factor_ expr_, {var_, min_, max_}] /; 
     FreeQ[factor, var] :> factor Integrate[expr, {var, min, max}])

and you can use it to pull out constants (or independent variables) and Mathematica will apply the rule above.

Regards,

Neil

Posted 11 months ago

Thank you, Neil.

My first trials with your recommended moveconst-function worked very well.

Regards Uwe

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