# Cannot evaluate TransformedDistribution for vectorial sum

I'm trying to obtain the distribution of the magnitude (length) of the vectorial sum of two vectors under a random (uniformly distributed) angle $\phi$. The sum is $\sqrt{x^2+y^2+2xy\cos(\phi)}$, where $x$ and $y$ are the lengths of the two vectors.

This is my code:

ModelTest = NormalDistribution[10, 5];
TailExp = ExponentialDistribution[0.2];
Sumf[x_, y_, phi_] = Sqrt[x*x + y*y + 2*x*y*Cos[phi]];
Trafo = TransformedDistribution[
Sumf[x, y, z], {
x \[Distributed] ModelTest,
y \[Distributed] TailExp,
z \[Distributed] UniformDistribution[{0, Pi}]
}
];

However, evaluating Trafo for a single point fails:

PDF[Trafo, x][5]

does not evaluate to a number.

I am rather new to Mathematica, so it's not clear to me whether there is a problem in my code or my approach in general?

Edit (on request I am adding more information on the final goal for which this setup is a mock-up):

I have a measured distribution of a physical quantity which I assume to be the vectorial sum of two random components. For one component (the bulk at low values) I have an analytical model. Now I need to find the other distribution (that is dominant in the tail). Unfortunately, it is not obvious what distribution that would be. (It is certainly not exponential because that is too steep, but it seemed to be simplest to use for the mock-up.) In the end I need to do a fit to determine the parameters of the tail distribution.

My intention was to use Mathematica to try a few different distributions for the tail to see what the overall distribution looks like and if it resembles the measurement. However, it seems to be simpler (and resonably fast) doing a naive numerical convolution instead.

• Shouldn't it be PDF[Trafo, 5] ? – chris Feb 15 '15 at 14:38
• I did try Summe = PDF[Trafo, x]; Summe2[5] (that's what I had originally) and PDF[Trafo,5], but none of them works. – fuenfundachtzig Feb 15 '15 at 14:56
• @fuenfundachtzig first part is certainly wrong. The latter ought to work unless Mathematica can't handle this configuration (TransformedDistribution fails on many of the less trivial transforms). – Sjoerd C. de Vries Feb 15 '15 at 22:27
• @SjoerdC.deVries: What do you mean by first part? Summe = PDF[Trafo, x]; Summe[5]? (Note that there is a typo in my previous comment. You don't mean that, do you?) – fuenfundachtzig Feb 16 '15 at 9:23
• @fuenfundachtzig I was referring to your comment and I noticed the typo, but that's not what I meant. your calculation boils down to PDF[Trafo, x][5] and that is incorrect. You want to have the PDF of trafo evaluated at x=5. That's done by PDF[Trafo, 5] as Chris said or by PDF[Trafo][5] and not by PDF[Trafo, x][5]. – Sjoerd C. de Vries Feb 16 '15 at 15:23

In order to get a feeling about what is going on here I suggest to perform a simulation.

Let

rx := RandomVariate[NormalDistribution[10, 5]];
ry := RandomVariate[ExponentialDistribution[0.2]];
rphi := \[Pi] Random[];

be the simulated variables, and

s := Module[{x = rx, y = ry, phi = rphi}, Sqrt[x^2 + y^2 + 2*x*y*Cos[phi]]];

the function whose distribution we are looking for.

Let's now calculate a sample of 10^5 numbers s:

ss = Array[s &, 10^5];

The histogram of ss gives the requested distribution

Histogram[ss]
(* 150216 _Histogram.jpg)

The resulting distribution resembles a Gamma Distribution.

Remark 1: I tried to perform the transformation analytically, and failed even for just one variable of the three. But this path should be followed more closely.

• Hm. Why would there be imaginary values for s? The argument of the sqrt function is always non-negative. – fuenfundachtzig Feb 16 '15 at 9:21
• Oh, I see one potential problem. Both x and y should be cutoff at 0 (i.e. also non-negative). How do I do that (for x)? But still, s should be non-negative even if x becomes negative... – fuenfundachtzig Feb 16 '15 at 9:27
• Oh, I made a mistake. I'll correct it immediately. – Dr. Wolfgang Hintze Feb 16 '15 at 11:14
• @fuenfundachtzig: you were right, hundertprozentig! ;-) – Dr. Wolfgang Hintze Feb 16 '15 at 11:42
• ;) Any other ideas? – fuenfundachtzig Feb 16 '15 at 16:56