Dependent Chi Square Distribution Random Variable

Question

If $X, Y, Z$ are I.I.D normally distributed random variable. Then what is the probability density function for the function given by $\max(\mid x-y\mid ^2,\mid y-z\mid ^2)$.

Answer 1

Comment: Maybe you can start with $X, Y, Z$ standard normal. The mean is not relevant, but you would need to adjust for $\sigma$ different from 1. Then $(X-Y)^2$ and $(Y-Z)^2$ have identical distributions that are related to $Chisq(df=1)$, but they are not independent. Of course, the depencence somewhat complicates finding the distribution of the maximum.

Some information in the following simulation may be useful.

 m = 10^5;  x=rnorm(m); y = rnorm(m); z = rnorm(m)
 u = (x-y)^2;  v = (y-z)^2
 mean(u); sd(u); mean(v); sd(v)
 ## 1.998461
 ## 2.836097
 ## 2.003054
 ## 2.837713
 cor(u,v)
 ## 0.2499633

 w = pmax(u,v)
 mean(w);  sd(w)
 ## 3.106980
 ## 3.356281