# Product of Gaussian and Rayleigh distributions gives what distribution?

• A
• Mishra
In summary, the conversation is about finding the distribution function for the product of two independent random variables, one following a non-zero-mean Gaussian distribution and the other a Rayleigh distribution. The math involved is complex and the density for the product of a zero-mean Gaussian and a Rayleigh is not applicable. Possible solutions include looking into copulas or using Monte Carlo simulation methods.
Mishra
TL;DR Summary
Finding out the distribution of the product of a Gaussian and a Rayleigh distributed random variables.
Hello,

I'm trying to find out the distribution function (cumulative or density) of the product of two independent random variables respectively following a non-zero-mean Gaussian and a Rayleigh distribution. The math is too intricate for me, I've found in the appendix of [Probability Distributions Involving Gaussian Random Variables - Simon, Marvin K.] the density for the product of a zero-mean Gaussian and a Rayleigh but this will not work for what I am trying to do.Would anyone have a reference that could help me ?

Thanks a lot for your help ! The answer to your question depends on the exact forms of the distributions that you are working with. Generally speaking, the product of two independent random variables will not have a closed form distribution and can be difficult to work with analytically. You may find useful results in the literature on copulas or related topics. For example, the book Copula Methods in Finance (McNeil et al., 2005) has several chapters devoted to the topic. Alternatively, you may need to resort to Monte Carlo simulation approaches that sample from the involved distributions and then compute the product of the sample values.

## 1. What is the product of Gaussian and Rayleigh distributions?

The product of Gaussian and Rayleigh distributions is a mathematical operation that combines the probability density functions of the two distributions to create a new distribution. It is calculated by multiplying the values of the two distributions at each point.

## 2. What is the significance of the product of Gaussian and Rayleigh distributions in scientific research?

The product of Gaussian and Rayleigh distributions is commonly used in signal processing, image processing, and communication systems to model the noise present in these systems. It is also used in various statistical analyses and machine learning algorithms.

## 3. What is the mathematical formula for the product of Gaussian and Rayleigh distributions?

The mathematical formula for the product of Gaussian and Rayleigh distributions is given by:
f(x) = (x/σ2) * e-(x2 + y2)/(2σ2)
where σ is the standard deviation of the Gaussian distribution and x and y are the values of the two distributions at each point.

## 4. How is the product of Gaussian and Rayleigh distributions related to the Central Limit Theorem?

The product of Gaussian and Rayleigh distributions is a special case of the Central Limit Theorem, which states that the sum of a large number of independent random variables will tend towards a Gaussian distribution. In this case, the product of two independent Gaussian and Rayleigh distributions results in a Gaussian distribution.

## 5. Can the product of Gaussian and Rayleigh distributions be used to model real-world phenomena?

Yes, the product of Gaussian and Rayleigh distributions can be used to model various real-world phenomena, such as the distribution of wind speeds, the intensity of earthquakes, and the strength of radio signals. It can also be used to model the noise present in electronic devices and systems.

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