# 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
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Posted 11 months ago
 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