I think integrating over sets is a useful thing to allow, but we
definitely need to be more careful about symbolic intervals. And given
that, it probably means that sympy.stats should just do normal
integrals, since it can make assumptions about symbolic intervals that
won't be present when passed to Integral.
Aaron Meruer
On Thu, Mar 26, 2015 at 6:54 PM, Francesco Bonazzi
<franz.bona...@gmail.com> wrote:
> Matthew... what do you think of the union of intervals as an alternative to
> the usual ranges in integrate/Integral?
>
> I suppose that you wrote the code outputting that integral, which currently
> does not work, and I want to make it work.
>
> I am undecided on whether to edit sympy.stats in order to give Integral( ...
> , (x, -oo, -1)) + Integral( ... , (x, 1, oo)) instead of Integral( ... , (x,
> Union(Interval(-oo, -1), Interval(1, oo)))).
>
> On the other hand, this alternative notation may be useful. Unfortunately it
> would require some algorithmic changes and I am a bit wary about a
> substantial edit of the integration algorithm.
>
> On Thursday, March 26, 2015 at 9:12:37 PM UTC+1, Matthew wrote:
>>
>> You don't need to square the random variable to compute the result. You
>> just need to integrate the pdf over x < -1 and x > 1
>>
>> On Thu, Mar 26, 2015 at 5:42 AM, Francesco Bonazzi <franz....@gmail.com>
>> wrote:
>>>
>>>
>>> Well, I was a bit surprised too, but the stats module apparently does so,
>>> as shown in this example:
>>>
>>> In [1]: from sympy.stats import *
>>>
>>> In [2]: var('sigma', positive=True)
>>> Out[2]: σ
>>>
>>> In [3]: N = Normal('X', mu, sigma)
>>>
>>> In [6]: P(N**2>1, evaluate=False)
>>> Out[6]:
>>> (-∞, -1) ∪ (1, ∞)
>>> ⌠
>>> ⎮ 2
>>> ⎮ -(z - μ)
>>> ⎮ ──────────
>>> ⎮ 2
>>> ⎮ ___ 2⋅σ
>>> ⎮ ╲╱ 2 ⋅ℯ
>>> ⎮ ───────────────── dz
>>> ⎮ ___
>>> ⎮ 2⋅╲╱ π ⋅σ
>>> ⌡
>>>
>>>
>>> In [7]: srepr(P(N**2>1, evaluate=False))
>>> Out[7]: "Integral(Mul(Rational(1, 2), Pow(Integer(2), Rational(1, 2)),
>>> Pow(pi, Rational(-1, 2)), Pow(Symbol('sigma'), Integer(-1)),
>>> exp(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('sigma'), Integer(-2)),
>>> Pow(Add(Dummy('z'), Mul(Integer(-1), Symbol('mu'))), Integer(2))))),
>>> Tuple(Dummy('z'), Union(Interval(-oo, Integer(-1), S.true, S.true),
>>> Interval(Integer(1), oo, S.true, S.true))))"
>>>
>>>
>>> Apart the fact that such an integral looks wrong to me, i.e. there is no
>>> account for the random variable being squared (or am I missing something?),
>>> it looks like SymPy is OK with intervals, but not with unions of intervals:
>>>
>>>
>>> https://github.com/sympy/sympy/blob/9242d31f6d31a1d9c3464264a5a6e61eab8acfb8/sympy/concrete/expr_with_limits.py#L37
>>>
>>> That's the point where an Interval gets parsed by the integration
>>> algorithm.
>>>
>>> I think it's an easy fix to add the processing for unions of intervals.
>>>
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>>
>>
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