What would be the PDF of the finite sum (i.e., the sum of N product terms) of the product of two independent but non-identically distributed Rayleigh random variables? Does someone know what the PDF would be (a paper, book, etc. where I can find it)?
The question can be formulated as
\sum_{n=1}^{N} { X_n Y_n },
where X_n and Y_n are independent but non-identically distributed Rayleigh random variables.
Is there a closed-form for this PDF?
Many thanks in advance!
We derive the exact probability density functions (pdf) and distribution functions (cdf) of a product of n independent Rayleigh distributed random variables. The case n=1 is the classical Rayleigh distribution, while n≥2 is the n-Rayleigh distribution that has recently attracted interest in wireless propagation research. The distribution functions are derived by using an inverse Mellin transform technique from statistics, and are given in terms of a special function of mathematical physics, the Meijer G-function. Series forms of the distribution function are also provided for n=3, 4, 5. We also derive a computationally simple moment-based estimator for the parameter occurring in the distribution, and evaluate its variance .
Dear Sajal Sasmal , I think you forgot to mention the paper this abstract is from. Are you referring to this paper: "The distribution of the product of independent Rayleigh random variables" by Jari Salo et. al.?
Dear Felipe,
I think it will be more expressive that you express your problem in a mathematical form. Representing the problem in a mathematical form is half the solution.
Best wishes
Dear Abdelhalim abdelnaby Zekry , as per your suggestion, I have added the mathematical form of this problem.
Dear Felipe,
I found some literature that can help you solve your problem.
- There is a closed form solution for the pdf of a summation of random Rayleigh variables: Article Accurate closed-form approximations to Rayleigh sum distribu...
- in addition there is a closed form solutions for a pdf pf a multiplication of Rayleigh variables: Article The Distribution of the Product of Independent Rayleigh Rand...
There is only one step if you combine the two solutions then you can get the required solution which is the sum of products of the Rayleigh random variable.
This may considered as a methodology for the solution.
Best wishes
Please read my paper
Generation of any probability density function from a Set of Equally Likely Random Variables
You will find
a good answer in it.
Dear Felipe,
any information about x_n, y_n?
have you a set of 2N parameters?
X_n and Y_n are Rayleigh distributed random variables. Additionally, 2N is also valid for the summation.
If N is sufficiently large (N>7), you may obtain useful approximate evaluations using Lidenberg–Levi central limit theorem - CLT (see e.g. G.Korn, T. Korn, Mathematical Handbook for Scientists and Engineers, in elder editions, point 18.6).
As I understand, you have the sum of N random products: SN = sum_(n=1,…,N)_(Xn*Yn), where Xn and Yn, (n=1,…,N) are independent random values, and each Xn and Yn are distributed according to own Rayleigh PDF. You did not write, but let us assume that both sequences are stationary (their PDF does not depend on n).
In this case (for the “large” N), distribution p(SN) of random sum SN is asymptotically normal, and its mean value “mean_SN” can be assessed by the relationship: mean_SN = sum_(n=1,…,N)_mean_(Xn*Yn) = sum_(n=1,…,N)_[mean_(Xn)*mean (Yn)] = N*mean_(Xn)*mean_(Yn)]^2. The variance of this distribution “var_SN” takes the value: var_SN = sum_(n=1,…,N)_var (Xn*Yn) = sum_(n=1,…,N)_[(Xn*Yn) - mean_(Xn*Yn)]^2 = N*{mean_(Xn*Yn)^2 – [mean_(Xn*Yn)]^2} =…
These, externally “terrible” relationships will be much simpler being written in MathType. Computation of one-dimensional values “mean_(Xn*Yn)” and “var_(Xn*Yn)” is not complex.
If Xn and Yn are not stationary, even correlated, the CLT may give more complex but useful approximations of p(SN).
Let you try (even if N ~ 5 ). It is interesting – will it give you any useful result? Wish you successes.
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