Re: a couple of gripes

*To*: mathgroup at smc.vnet.net*Subject*: [mg29772] Re: [mg29751] a couple of gripes*From*: "Mark Harder" <harderm at ucs.orst.edu>*Date*: Sun, 8 Jul 2001 01:00:16 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Joshua, In response to: >When I evaluate >Integrate[f[x]*If[x==y,1,0],{x,-Infinity,Infinity}] >I don't get >f[y]. That's because Mathematica doesn't recognize that your If[] is equivalent to DiracDelta[x-y]. In general (see numerous postings in the MathGroup archive), one should use the Generalized Function construction equivalent to the "black box" If[] function. Observe: In[967]:=Clear[f] Integrate[f[x]*DiracDelta[x - y ], {x, -Infinity, Infinity}] Out[968]= f[y] and re: >When I evaluate >PDF[NormalDistribution[x,0],x] >I don't get >1. I believe this is an incorrect expectation as follows: In[974]:= Limit[PDF[NormalDistribution[mu, a], x], a -> 0] Out[974]= Indeterminate The NormalDistribution is not defined for sigma<=0, and in the limit sigma->Infinity, I think the PDF of the NormalDistribution at x=mu goes infinite. However, the *area* under the PDF over its infinite domain is 1: In[988]:=Simplify[Integrate[PDF[NormalDistribution[mu, a], x], {x, -Infinity, Infinity}], {a \[Element] Reals, a > 0 }] Out[988]=1 For an example of this behavior, see 3.5.12 in The MathematicaBook, concerning Generalized Functions. -mark harder -----Original Message----- From: Joshua A. Solomon <J.A.Solomon at city.ac.uk> To: mathgroup at smc.vnet.net Subject: [mg29772] [mg29751] a couple of gripes >When I evaluate >Integrate[f[x]*If[x==y,1,0],{x,-Infinity,Infinity}] >I don't get >f[y]. > >When I evaluate >PDF[NormalDistribution[x,0],x] >I don't get >1. > >js >-- >Joshua A. Solomon >Department of Optometry and Visual Science >City University >London EC1V 0HB >Voice: (44) 20 7040 0192 >Secretary/Fax: (44) 20 7040 0182 >J.A.Solomon at city.ac.uk >http://www.staff.city.ac.uk/~solomon > >