# How to prove that a gaussian rv multiplied by a sine also guassian ?

• dexterdev
In summary, when considering a random process with an independent variable of time, an ensemble of random waves of the same experiment is taken into account. At a specific time, if the sinusoid is a constant and multiplied by a gaussian random variable, the resulting random variable will also be gaussian. However, to prove that B is a gaussian process, it is important to consider linear functionals applied to the sample function. This will result in a normally distributed outcome.

#### dexterdev

How to prove that a gaussian random variable multiplied by a deterministic sinusoid also results in a random variable with gaussian pdf?

Suppose here A is a guassian random variable and B is given as below where fc is the frequency of sinusoid and t is the time.

B=A∗cos(2π fc t). Is random variable B also gaussian. If so, how to prove it?

This may be a foolish question and/ or direct one . But please help me, I cannot figure it out.

It isn't a foolish question, but it is an ambiguous question. When you take random samples from B, how are you picking the times at which to take these samples? For example, are you sampling B every 10 seconds? Or are you selecting the times to take the samples from some probability distribution?

B = B(t) is a stochastic process, which is Gaussian for each t.

Stephen Tashi said:
It isn't a foolish question, but it is an ambiguous question. When you take random samples from B, how are you picking the times at which to take these samples? For example, are you sampling B every 10 seconds? Or are you selecting the times to take the samples from some probability distribution?

Hi Sir, I believe that I have figured it now. Please correct me if I am wrong. When considering random processes (independent variable is time t, right?) we consider an ensemble of random waves (of same experiment). So at a particular 't = T' sinusoid is a constant and a constant times gaussian rv is again gaussian. Is this right?

dexterdev said:
So at a particular 't = T' sinusoid is a constant and a constant times gaussian rv is again gaussian. Is this right?

That is correct, as mathman pointed out. But if you are trying to write a proof that B is a gaussian process you should say more. For example, in the Wikipedia article on Gaussian Process, note the line:

More accurately, any linear functional applied to the sample function Xt will give a normally distributed result

Thankyou Sir

## 1. What is a Gaussian random variable?

A Gaussian random variable (also known as a normal random variable) is a type of probability distribution that is commonly used to model natural phenomena. It is characterized by its bell-shaped curve, with the majority of the data clustered around the mean and symmetrically distributed on either side.

## 2. How do you prove that a Gaussian random variable multiplied by a sine is also Gaussian?

The proof involves using the properties of the Fourier transform and the fact that the product of two Gaussian functions is also a Gaussian function. This can be shown mathematically by taking the Fourier transform of the product of a Gaussian and a sine function, and then using the convolution theorem to show that the resulting function is also a Gaussian.

## 3. What is the significance of proving that a Gaussian random variable multiplied by a sine is Gaussian?

This proof has practical applications in signal processing and statistics, where it can be used to model and analyze various phenomena. It also demonstrates the versatility and importance of the Gaussian distribution in mathematical and scientific fields.

## 4. Are there any limitations to this proof?

As with any mathematical proof, there may be certain assumptions or conditions that need to be met for it to be valid. Additionally, the proof may not hold true for all possible variations or combinations of Gaussian and sine functions.

## 5. Are there other methods for proving this statement?

Yes, there are other methods that can be used to prove this statement, such as using characteristic functions or moment-generating functions. However, the Fourier transform approach is one of the most widely used and intuitive methods for proving this statement.