Which is the product of two independent Gaussian random variables?

Which is the product of two independent Gaussian random variables?

Since they have the same variance, X − Y and X + Y are independent Gaussian random variables. Put Z := X 2 − Y 2 2 = X − Y 2 X + Y 2. Then Z is the product of two independent Gaussian, but the characteristic function of Z is φ Z (t) = 1 1 + t 2, which is not the characteristic function of a Gaussian.

Which is the formula for multivariate Gaussian density?

To get an intuition for what a multivariate Gaussian is, consider the simple case where n = 2, and where the covariance matrix Σ is diagonal, i.e., x = x1 x2 µ = µ1 µ2 Σ = σ2 1 0 0 σ2 2 In this case, the multivariate Gaussian density has the form, p(x;µ,Σ) = 1 2π σ2 1 0 0 σ2 2 1/2 exp − 1 2 x1 −µ1 x2 −µ2 T σ2 1 0 0 σ2 2 −1 x1 −µ1 x2 −µ2 ! = 1 2π(σ2

Are there any cases where X and Y are non-independent?

$\\begingroup$This proves that there are cases where X and Y are independent and XY isn’t Gaussian, and cases where X and Y are non-independent and XY isn’t Gaussian. However, there is also the question of whether cases exist in which XY is Gaussian (which would mean that X and Y would have to be non-independent).

Is the product of two PDFs a Gaussian?

(I think a few real world examples would clear things up for me) The product of the PDFs of two random variables X and Y will give the joint distribution of the vector-valued random variable ( X, Y) in the case that X and Y are independent.

Which is the posterior probability of two Gaussian PDFs?

The measurement model tells us that P [ X ~ ∣ X] is Gaussian, in particular P [ X ~ ∣ X] = N [ Σ ϵ, X]. Since the product of two Gaussians is a Gaussian, the posterior probability is Gaussian.

Is the product of two random variables Bivariate?

The product of the PDFs of two random variables X and Y will give the joint distribution of the vector-valued random variable ( X, Y) in the case that X and Y are independent. Therefore, if X and Y are normally distributed independent random variables, the product of their PDFs is bivariate normal with zero correlation.