This should be a rather simple question, and I think I am just not understanding the error.
I have a function say f(x,y)= x*y^3*normal(y,1,2) where normal() is the normal pdf equation (x,mean,standard deviation).
So I want to integrate across the pdf and then plot x. What I thought I would do was:
g = f.integral(y,0,Infinity)
plot(g,-10,10)
But that give the error: ValueError: free variable: x
What am I missing?
Thanks; I don't quite understand what's broken, but I think I have a workaround.
sage: normal(x,av,sd)=((1/(sd*sqrt(2*pi)))*exp(-(x-av)^2/(2*sd^2))) sage: f(x,y)= x*y^3*normal(y,1,2) sage: g = f.integrate(y,-2,2) sage: g (x, y) |--> 1/4*sqrt(2)*x*integrate(y^3*e^(-1/8*(y - 1)^2), y, -2, 2)/sqrt(pi)
(note that sage seems to think g is a function of x and y, even though it shouldn't be)
sage: h(x) = f.integrate(y,-2,2) sage: h x |--> 1/4*sqrt(2)*x*integrate(y^3*e^(-1/8*(y - 1)^2), y, -2, 2)/sqrt(pi)
(this is looking better, but still doesn't work . . . the problem seems to be the integrate command in the definition of h)
sage: plot(h,0,5) ... ValueError: free variable: y
(the first way I could think of to get Sage to evaluate the integral was to use the numerical_approx method; maybe there's a better way, but this works :)
sage: k = lambda x: h(x).numerical_approx() sage: plot(k,0,5)
(graph is shown!)
Note that, unfortunately, the following does not work:
k(x) = h(x).numerical_approx()
...
TypeError: cannot evaluate symbolic expression numerically
Unless someone knows better, I'm inclined to think this should be filed as a Trac ticket.
EDIT: after writing all this, I think the problem is the "symbolic expression" integrate(...). The following works as expected:
sage: k(x) = 1/4*sqrt(2)*x*(integrate(y^3*e^(-1/8*(y - 1)^2), y, -2,2).numerical_approx())/sqrt(pi)
sage: plot(k,0,5)
To some extent this is a design decision - what variables should an integral of a function have as variables? (Arguments is probably the better word.) For indefinite ones, and derivatives, we keep them, and so since this had a symbolic evaluation, it's still a function of both variables... hmmm...
I agree, it must be a problem with normal. The following works as expected
sage: f(x,y)= x*y^3 sage: f (x, y) |--> x*y^3 sage: g = f.integral(y,0,7) sage: g (x, y) |--> 2401/4*x sage: plot(g,-5,5)
Where as this returns the error you've seen:
sage: f(x,y,z)= x*y^3+z
sage: g = f.integral(y,0,7)
sage: plot(g,-5,5)
...
ValueError: free variable: z
If you try to simply integrate the normal function, do you get a number, or another function of x?
Asked: 2010-08-26 13:09:11 +0100
Seen: 5,807 times
Last updated: Aug 26 '10
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Is this a case of not declaring symbolic variables with var()?
ValueError is documented in python as an exception thrown when the domain of the passed argument doesn't match the function domain if I read the docs correctly.
I'd like to try to help, but I don't find `normal` as a builtin function. Do you import it, or have you defined it yourself? The error comes from Sage trying to use `fast_float` to evaluate your function and apparently finding two variables somehow. Maybe post your whole session?
See below.