@sand15 

Thank you again for your attention.

@sand15 

This is helpful.  Thank you.

@acer Do you have any suggestions on what I should do?

@acer 

Yes -- combining the individual PDFs into a single, sum of the three individual PDFs is what I seek.  Thank you.

@sand15 

The use of logarithms had never crossed my mind -- thank you.

@acer 

The assumption makes sense.  Thank you.

@sand15 

Thank you again for all of your help.  I am still unable to get a closed-form PDF for the random variable that is the absolute difference between two random variables.

When I look at your work, I understand what is happening.  I just don't know how to capture the exact PDF in symbolic form.  I have attached a revised version of your work (with some details omitted that I don't need).  The "unapply" functionality confuses me -- perhaps that is part of the problem.

I have also included a simple worksheet where I capture an estimate of the PDF via "twice the convolution."  I just can't capture the symbolic solution of the PDF.

Thanks again.

Download DifficultConvolution.mw

Download Sand15Help.mw

@sand15 

Just a simple omission -- I apologize.  I appreciate all of your recent help.

@sand15 

Thank you for your help.  I now understand how 2*Convolution of X and Y can result in the density function of Z, where Z = |X - Y|.  In fact, a few months ago, I wrestled with this same issue when considering Z = |X - Y| when X and Y were both uniformly distributed on the [0,1] interval.

Of course, X and Y both being uniformly distributed on [0,1] is more complicated than my current dilemma.

I looked at the worksheet you provided.  Some of the things you asked for I did not understand.  I pruned away some of the items that were not necessary for me.

I understand this is to simulate values of Z:

S__Z := Sample(Z, 10^6):

plots:-display(
  Histogram(S__Z, transparency=0.7, minbins=100, style=polygon)
  , plot(f__U(u), u=0..2, color=red, thickness=2, legend=typeset(:-PDF('Z')))
)

I did not understand the section below, but I think it is a numerical estimate.

F__U := proc(u)
  option remember:
  local F1;
  F1 := 2*evalf(Int(op([2, 2], f__U(t)), t=0..1)):
  if _passed::numeric then 
    if u <= 1 then
      2*evalf(Int(op([2, 2], f__U(t)), t=0..u))
    elif u <= 2 then
      F1 + 2*evalf(Int(Re(op([2, 4], f__U(t))), t=1..u, method = _d01ajc))
    end if:
  else
    procname(_passed)
  end if:
end proc:

plot([seq([u, F__U(u)], u=0..2, 0.1)], color=red, thickness=2, legend=typeset(:-CDF('Z')), gridlines=true)
Error, (in F__U) improper op or subscript selector

I (think I) understand the below to be the exact Mean of Z, which is great.

:-Mean('Z') = Mean(Z);

But when I ask for the PDF of Z, I get a bunch of garbage.  But I do get the expected result when I entered plot(f__U(u), u = 0 .. 2).

My modification of your work is included in PDFDerivation.mw.

As such, I moved ahead with convolving X and Y to see what would happen.  Unfortunately, I was unable to integrate the product of the X and Y.  This work is also attached (Convolution.mw).  I "shifted" the t variable to the right so that t is always positive.

Download PDFDerivation.mw
 

Download Convolution.mw

@sand15 

Thank you for your reply.  

Y1 and Y2 DO have the same distribution.  I was incorrect when I stated my simulated result of |Y1 - Y2| takes on a logrithmic form.  I had the same result as your simulated PDF.

I need to find the exact PDF of |Y1 - Y2|.

I know how to sum PDFs to find the new PDF.  I know how to multiply PDFs to find the new PDF.  But the absolute difference between two PDFs into a new PDF is something new to me.

@mmcdara 

Thank you -- this is helpful.

@acer 

Thank you.  I do not need the symbolic solution -- I was hoping it would be "tidier." 

The evalf solution was just a check to see if the result was as expected when compared to the simulated result.

I am good.  Thanks again.

@mmcdara 

Thank you for your help with this issue.  I have had success approximating PDFs with the Pearson Distribution.  This is what you suggested:  PDF_analytic_approximation.mw

I have recently attempted to approximate another random variable that I wasn't able to simulate via Maple.  I had to simulate the data via Java.  I then imported the simulated data into Maple with the intent of approximating the PDF via the Pearson Distribution.  I was successful in bringing the data into Maple via the Excel import and create a histogram.  I ran into problems when I attempted to approximate the Pearson disbtibution in Maple.  I think the "data type" is the problem.  The data was imported as a matrix.  I then tried to convert the data to Vector and Array form, but without success.

The data and my attempt at approximating the Pearson Distribution as as follows:  MyData.xlsx, MyApproximationAttempt.mw.

@mmcdara thank you.  I had no idea these integration tools are available.

This brings me to a follow-up question.  This density function:  fw:= 2*sqrt(1-t²)/Pi...can it be classified as a Random Variable?

Thank you, acer.  I did not realize that the PDF function could be followed by the "assuming" keyword.  

My guess would be that my omitting this ended up confusing Maple.