# Integrate a product of functions with random variables

Assume there is a function f($$x_1$$,...$$x_n$$,$$Y$$) and I seek for the integral following on an area $$[0,1]^m$$:

where $$Y_i$$ are the parameter I generate randomly for each function. The total number of the functions $$n$$ is 200 ~ 2000, which means I cannot type directly into Mathematical to calculate the integral value. Is there some way to realize this kind of integral?Approximate value is acceptable. Thank you!!

Does this work?

Integrate[
Product[f[x1, x2, x3, RandomVariate[UniformDistribution[]]], {j, n}],
{x1, 0, 1}, {x2, 0, 1}, {x3, 0, 1}]

Test:

n=6;
f[xx_, yy_, zz_, ww_] := xx + yy + zz + ww;

Integrate[
Product[f[x1, x2, x3, RandomVariate[UniformDistribution[]]], {j, n}],
{x1, 0, 1}, {x2, 0, 1}, {x3, 0, 1}]

(*

163.454

*)

• Interesting!!! Let me have a try – 0o0o0o0 Jan 12 at 20:49
• This can solve half of the question, but then I have the following error during NIntegral: .... has evaluated to non-numerical values for all sampling points in the region with boundaries...... – 0o0o0o0 Jan 14 at 0:54