MaplePrimes - answers and comments on Question, truncated normal distribution

TruncNormal

Markiyan Hirnyk — Fri, 22 Jul 2011 09:42:12 Z

Folowing a recent post http://www.mapleprimes.com/questions/123721-Statistics-How-To-Create-Parametric-Distribution , one can create a distribution in such a way:

Download TruncNormal.mw Look at ?Statistics[Distribution] for more details.

PS. The command

DensityPlot(TruncNormal(0, 1, 2, 3), view = [-2 .. 6, -1 .. 4])

is better as we see a larger part of the plot.

Regarding the 2nd question, if F has a normal

longrob — Fri, 22 Jul 2011 13:43:09 Z

Regarding the 2nd question, if F has a normal distribution truncated at [a,b] , it's PDF is

pdf_tn:=PDF(TruncNormal(mu,sigma,a,b),x);

         /                   /          /          2\\//                
         |                   | (1/2)    |  (x - mu) || |    (1/2)       
piecewise|a          |                   |          |         2 || \                
         \                   \          \  2 sigma  //                  

  /     /          (1/2)\        /          (1/2)\\\   \
  |1    |(b - mu) 2     |   1    |(a - mu) 2     |||   |
  |- erf|---------------| - - erf|---------------|||, 0|
  \2    \    2 sigma    /   2    \    2 sigma    ///   |
                                                       /

Then, the PDF of a normal distribution with mean mu+t, truncated at [a,b] is

subs(x=y-t,pdf_tn);
         /                           /          /              2\
         |                           | (1/2)    |  (y - t - mu) |
piecewise|a          |                           |          |           2   |
         \                           \          \    2 sigma    /

  \//                /     /          (1/2)\
  | |    (1/2)       |1    |(b - mu) 2     |
  | |2 Pi      sigma |- erf|---------------|
  | \                \2    \    2 sigma    /
  /                                         

          /          (1/2)\\\   \
     1    |(a - mu) 2     |||   |
   - - erf|---------------|||, 0|
     2    \    2 sigma    ///   |
                                /

Note that, formally, to compute this PDF, we need to divide by abs(diff(y(x),x)) 
which is just 1 in this case, so I omitted it.

Thanks a lot! This works for me!

kelvinlee — Fri, 22 Jul 2011 22:01:33 Z

Thanks a lot! This works for me!

a general case

kelvinlee — Fri, 22 Jul 2011 22:00:00 Z

Thanks a lot for your help! For the PDF returned by the substitution did you mean the function was for the normal distribution (mu+t, delta) truncated at [a+t, b+t] instead of at [a, b]?

Also could you explain a little bit about the math principal you used there? What if the expression X + Y where Y is a untruncated normal distribution (mu_2, delta_2), and X is still a truncated distribution (mu, delta) at [a, b], is it still possible to derive the corresponding truncated normal distribution using the same rule you mentioned? Thanks!

Yes, the truncation is effectively shifted

longrob — Sat, 23 Jul 2011 01:39:27 Z

Yes, the truncation is effectively shifted by t. To see the attached document with 2 plots

The mathematical principle is the concept of "function of a random variable" - I've appended it, as I understand it, briefly in the document also. I'm not sure of the best way to present "nice format" math in maple as this is the first time I tried it in a document. Hope it's readable, but if you just google for "function of a random variable" you should find plenty of references.

Edit1: Note that f(x) and f'(x) are hard to distinguish above :(
Edit2: f must be monotonic for this to work, otherwise it doesn't have a unique inverse. If f is not monotonic, then the usual way to proceed is to split it pieces that are monotonic

Download TruncNormal1.mw

Plot missing

longrob — Sat, 23 Jul 2011 01:43:19 Z

I don't know why the 2nd plot is missing on the above post. I'll try it again with a classic mode worksheet

>	restart; with(Statistics):

>	S := piecewise(A <= t and t <= B, PDF(NormalDistribution(Mu, Sigma), t)/(CDF(NormalDistribution(Mu, Sigma), B)-CDF(NormalDistribution(Mu, Sigma), A)), 0):

>	P:=proc(mu,sigma,a,b)Distribution(PDF = unapply(eval(S, [Sigma = sigma, Mu = mu, A=a,B=b]), t)) end proc:

>	TruncNormal:=(mu,sigma,a,b)->RandomVariable(P(mu,sigma,a,b)):

>	pdf_X:=PDF(TruncNormal(mu,sigma,a,b),x);

(1)

>	#if y = x + t then

>	pdf_Y:=subs(x=y-t,pdf_X);

(2)

>	# plot the pdf of the N(mu,sigma), truncated at [2,3]

>	plot(eval(pdf_X,{mu=0,sigma=1,a=2,b=3}), view = [-1 .. 5, -1 .. 3]);

>	# plot the pdf of N(mu+1,sigma), truncated at [2,3]

>	plot(eval(pdf_Y,{mu=0,sigma=1,a=2,b=3,t=1}), view = [-1 .. 5, -1 .. 3]);

Download TruncNormal1-1.mw