More Integrate woes
- To: mathgroup at smc.vnet.net
- Subject: [mg9584] More Integrate woes
- From: NOHAMcrose at c2.telstra-mm.net.au (Colin Rose)
- Date: Thu, 13 Nov 1997 01:40:48 -0500
- Organization: Theoretical Research Institute
- Sender: owner-wri-mathgroup at wolfram.com
In any statistical setting, calculation of the Gaussian distribution
function is extremely important. Under Mathematica v3, this is
something of a disaster.
To summarise the problem:
Under v3:
________
Expressions such as:
aa = Integrate[Exp[-x^2],{x,-Infinity,y},
GenerateConditions->False]
return output *of form*:
1 - Erf[Sqrt[y^2]]
Our user now seeks numerical output. S/he enters:
(aa /. y -> 3) == (aa/. y -> -3)
True
This is clearly FALSE.
BY CONTRAST, under v2.2:
_______________________
aa = Integrate[Exp[-x^2],{x,-Infinity,y}]
returned output of form:
1 + Erf[y]
which is correct for all real y. No problem.
________
The problem with v3 here is not only that setting
GenerateConditions->False
yields incorrect results for one of the most important and common
integration problems, but that even if we set
GenerateConditions->True
the resulting conditional statement treats the distribution as if it is
asymmetrical, when the normal distribution itself is clearly
symmetrical. For instance:
aa = Integrate[Exp[-x^2],{x,-Infinity,y}, GenerateConditions->True]
returns
If[ y < 0,
-(1/2)*Sqrt[Pi]*(-1 + Erf[Sqrt[y^2]]) ,
Integrate[E^(-x^2), {x, -Infinity, y}]
]
Choosing y < 0 as opposed to y > 0 as a basis for conditional output is
completely arbitrary, and precisely the cause of the
GenerateConditions->False bug.
It would be great to hear that this will be fixed in v3.1.
[ Co-author Murray Smith discovered these oddities
and has been cursing wolfies ever since. ]
Cheerio
Colin
--
Colin Rose
tr(I) - Theoretical Research Institute
______________________________________ NOHAMcrose at c2.telstra-mm.net.au
http://www.usyd.edu.au/su/tri/