Gaussian Random Variables Question

[chingkui]

How do you show that a linear combination of two Gaussian Random Variables is again Gaussian?

 

[Hurkyl]

By working out what the actual distribution of the linear combination, I'd presume. This homework?

 

[tacman]

To show that a linear combination of two Gaussian random variables is again Gaussian, we can use the fact that the sum of two independent Gaussian random variables is also Gaussian.

Let X and Y be two independent Gaussian random variables with means μX and μY, and variances σX^2 and σY^2 respectively.

Now, let Z = aX + bY, where a and b are constants.

We can find the mean and variance of Z as follows:

E[Z] = E[aX + bY] = aE[X] + bE[Y] = aμX + bμY

Var(Z) = Var(aX + bY) = a^2Var(X) + b^2Var(Y) = a^2σX^2 + b^2σY^2

Since X and Y are independent, their joint distribution is also Gaussian. Therefore, the joint probability density function of X and Y can be written as:

fXY(x,y) = (1/2πσXσY)exp[-((x-μX)^2/2σX^2) - ((y-μY)^2/2σY^2)]

Now, using the change of variables method, we can find the probability density function of Z as follows:

fZ(z) = ∫∫fXY(x,y)dxdy

= ∫∫(1/2πσXσY)exp[-((x-μX)^2/2σX^2) - ((y-μY)^2/2σY^2)]dxdy

= (1/2πσXσY)exp[-((z-aμX)^2/2a^2σX^2) - ((z-bμY)^2/2b^2σY^2)]

This is the probability density function of a Gaussian random variable with mean aμX + bμY and variance a^2σX^2 + b^2σY^2, which is the same as the mean and variance we found for Z earlier.

Therefore, Z is a Gaussian random variable and a linear combination of two Gaussian random variables is again Gaussian.