For Z distributed on R^3 with Z1, Z2, Z3 IID Z ~ exp(1)
I have X = (Z2 + Z3)/(Z1 + Z2 + Z3)
How can i determine the distribution of X and obtain its mean and variance?
I was thinking perhaps I need to do something in terms of gamma because standard exponential is z ~ exp(1) iff z ~ G(1) and chi-square possibly???
plz someone can help me??
Mar 16th 2010, 04:31 PM
As you said, recall that exp(1) ~ Gamma(1)
Since X+Y ~ Gamma(a+b) if X ~ Gamma(a) and Y ~ Gamma(b), it follows that Z2+Z3 ~ Gamma(2)
Now there's this thing that you should know : if X and Y are independent, and X ~ Gamma(a) and Y ~ Gamma(b), then X/(X+Y) ~ Beta(a,b)
Hence, (Z2+Z3)/(Z1+[Z2+Z3]) follows a Beta distribution with parameters (2,1), because Z2+Z3 follows a Gamma(2) and Z1 follows a Gamma(1)