Re: piecewice pdf, problems with cdf
- To: mathgroup at smc.vnet.net
- Subject: [mg105413] Re: [mg105365] piecewice pdf, problems with cdf
- From: michael partensky <partensky at gmail.com>
- Date: Tue, 1 Dec 2009 04:16:52 -0500 (EST)
- References: <200911291012.FAA16385@smc.vnet.net>
On Sun, Nov 29, 2009 at 12:03 PM, <danl at wolfram.com> wrote:
> > Hi! Teaching the continuous distributions, I needed to introduce the
> > piecewise functions.
> > Here is the example that did not work well:
> >
> > In[56]:= f1[x_] /; 0 < x <= 3 := 1/9 x ^2;
> > f1[x_] := 0;
> >
> > Plot[f1[x],{x,-1,4}] works fine. However, the results for cdf are
> > ambiguous
> > In[57]:= cdf[x_] := Integrate[f1[v], {v, -\[Infinity], x}]
> >
> > In[59]:= cdf[1]
> > Out[59]= 0
> >
> > I thought that may be the second definition (for some reason) overwrote
> > the
> > first, but apparently this was not the case.
> >
> > Then I tried using Which,
> >
> > f1[x_] := Which[0 < x <= 3, x^2/9, x <= 0 || x > 3, 0];
> >
> > Plot[f2[x], {x, -1, 4}] worked fine.
> >
> > However, Plotting CDF is very slow.
> >
> > What is the reason for the first error and how to accelerate (compile?)
> > the
> > second?
> >
> > Thanks
> > Michael
> >
> > PS: I was aware about the issues with the derivatives of Piecewise
> > functions, but expected integration to be safe. What did i do wrong?
> > [...,]
>
> You did not use Piecewise. Those other methods do not play well with
> Integrate. Actually this has been discussed in this forum a couple of
> times in the past four months (though those discussions had a way of going
> off-line, and taking me with them).
>
> Anyway, use Piecewise; this is the sort of thing it is designed to handle.
>
> f1[x_] := Piecewise[{{x^2/9, 0 < x <= 3}}, 0]
>
> This will work fine. To make the cdf much faster, use Set rather than
> SetDelayed, and let Integrate know x is real valued.
>
> In[100]:= cdf[x_] = Integrate[f1[v], {v, -Infinity, x},
> Assumptions -> Element[x, Reals]]
>
> Out[100]= Piecewise[{{1, x > 3}, {x^3/27,
> Inequality[0, Less, x, LessEqual,
> 3]}}, 0]
>
> Daniel Lichtblau
> Wolfram Research
>
Thanks, Daniel.
Using 'Set' works great indeed!
Best
Michael