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Re: piecewice pdf, problems with cdf

  • To: mathgroup at smc.vnet.net
  • Subject: [mg105378] Re: [mg105365] piecewice pdf, problems with cdf
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Mon, 30 Nov 2009 06:11:06 -0500 (EST)
  • Reply-to: hanlonr at cox.net

pdf[x_] = Piecewise[{{x^2/9, 0 < x <= 3}}];

cdf[x_] = 
 Assuming[{Element[x, Reals]}, Integrate[pdf[t], {t, -Infinity, x}]]

Piecewise[{{1, x > 3}, 
     {x^3/27, Inequality[0, Less, x, 
         LessEqual, 3]}}]

cdf[1]

1/27

Plot[{pdf[x], cdf[x]}, {x, -1, 4},
 Frame -> True, Axes -> False]


Bob Hanlon

---- michael partensky <partensky at gmail.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?




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