Hi!
On 2017-10-14, Santanu Sarkar <[email protected]> wrote:
> In Sage, is it possible to find a such that
>
> \int_{a}^{\infty} e^(-x^2/2) dx=2^(-20)
Yes.
Probably you want to see how.
First, define a variable `a`. I don't know if one really needs
to declare its domain to solve the problem, but when one does,
the computation works:
sage: var('a', domain='real')
a
Compute the integral (which relies on the error function):
sage: F = (e^(-x^2/2)).integral(x,a,infinity)
sage: F
-1/2*sqrt(2)*sqrt(pi)*erf(1/2*sqrt(2)*a) + 1/2*sqrt(2)*sqrt(pi)
Solve for `a`. I want the solution to be given as a list of
dictionaries:
sage: LD = solve(F==2^(-20), a, solution_dict=True)
The solution is unique and is given in terms of the inverse of
the error function:
sage: len(LD)
1
sage: LD[0][a]
sqrt(2)*erfinv(-1/1048576*sqrt(2)/sqrt(pi) + 1)
You can evaluate it numerically, and then verify it's correct:
sage: a_num = RR(LD[0][a]); a_num
4.94513221771782
sage: RR((e^(-x^2/2)).integral(x,a_num,infinity)) - 2.^-20
0.000000000000000
Regards,
Simon
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