Let X and Y be jointly Gaussian random variables with means µX and µY and standa

Question

Let X and Y be jointly Gaussian random variables with means µX and µY and standard deviations X and Y respectively, and correlation coefficient XY . Define new random variables U1 and U2 by the transformation.

U1 = (X µX)/X + (Y µY )/Y

U2 = (X µX)/X (Y µY )/Y

Show that U1 and U2 are Gaussian random variables with zero means and zero correlation coefficient.

How would you modify the transformation to obtain two new independent Gaussian random variables V1 and V2 with zero means and unit variances?

Explanation / Answer

Question in probability & statistic

Applying the equation two-dimensional to Gaussian distribution.

(1)

Making transformed variables

U1 = (X µX)/X + (Y µY )/Y

U2 = (X µX)/X (Y µY )/Y

If we sum U1 + U2 is equal to

And if we subtract U1 –U2 is equal to :

Substituting these last two expression in (1)

We have

Define new random variables U1 and U2 by the transformation.

*

Show that U1 and U2 are Gaussian random variables with zero means and zero correlation coefficient.

If µX and µY are zero X and y are 1 and is zero then the distribution is called standard two-dimensional normal density function and is:

   (2)

U1 = (X µX)/X + (Y µY )/Y

U2 = (X µX)/X (Y µY )/Y

Then

U1 =

U2 =

If so,

U1 + U2 = 2X …….. X =

U1 - U2 = 2Y …….. Y =

Substituting these last two expression in (2)

We have:

*

Then simplifying:

*

This equation match with 2, and It is demonstrated that U1 and U2 are a Gaussian random variables with zero means and zero correlation coefficient.

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