# Gaussian random variable joint density with discrete pdf

Hi all,

I am having trouble with the concept of joint pdf's. For example - a set Z1,Z2,...ZN are each gaussian rv.

Let Z1~N(0,1), let X be +1 or -1 each with probability 0.5. Z2=Z1X1, so Z2 is ~N(0,1).

(I assume this to be As Z2 is just Z1 multiplied by a simple factor, an instance of X, either +1 or -1.)

I am having trouble understanding how the joint pdf pZ1Z2(z1,z2) is impulsive on the diagnols where z2=+/-z1 and is zero elsewhere.

I can understand how the joint pdf of for example pZ1X(z1,x) would be zero but impulsive on the diagnols but not when the joint pdf is made from the two gaussians as described - - advice appreciated

You have it backward. The joint PDF of z1 and x is never zero, at least not for values of either x or z1 that can occur. They're independent (I assume, although you didn't explicitly state this), so the joint PDF for z1 and x is just the product of the individual PDFs. (I'm ignoring the fact that x doesn't really have a well-behaved PDF.)

The is not the case for z1 and z2. From the definition, z2 is always either z1 or -z1. There are no other possibilities. So, for instance, although z1 = 0.5 is possible and z2 = 1 is possible, (z1,z2) = (0.5, 1) is impossible.

Thanks - I think I am a little closer to understanding,

As Z2=Z1X1, I was assuming that Z2's pdf was the joint pdf of Z1 and X, I take it that this is incorrect?

As Z2 can only equal + or - z1, this is what makes it impulisve at the points where Z2=Z1. Im just having a little dificilty visualizing the joint pdf (z1,z2), paticulalry the distribution of z2 alone.

As Z2=Z1X1, I was assuming that Z2's pdf was the joint pdf of Z1 and X, I take it that this is incorrect?
That is incorrect. z2 is a standard normal variable, so it has a nice ordinary bell-shaped PDF.

This is where Im having difficulty, if it is a standard bell curve then how can it take on only values of +z1 and -z1 when z1 itself is also a standard bell curve - Im having trouble visualizing a bell curve that contains all values of + and - another bell curve

The unconditional distribution of z2 is just a bell curve. The conditional distribution of z2, if you know z1, is a sum of two delta functions,

$$\frac{1}{2}\delta(z_2-z_1)+\frac{1}{2}\delta(z_2+z_1)$$

The joint PDF for z1 and z2 looks something like the attached plot, except that I broadened the ridges a bit to make them plottable. In reality, they would have width zero and infinite height.

#### Attachments

• normal_cross.png
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Thats great - thanks very much for making things clear

It's better to avoid pdfs when working with dependent random variables or continuous-discrete mixtures and the delta function approach can give paradoxical results if you're not extremely careful, however the joint cdf is always defined and for this example it is

F(x1,x2) = {(1/2)*N(xmin) if xmin<0<xmax<(-xmin) or N(xmin) otherwise}

where xmin=min(x1,x2), xmax=max(x1,x2) and N is the normal cdf.