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

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.

## Answers and Replies

Stephen Tashi
Science Advisor
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?

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

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?

Stephen Tashi
Science Advisor
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