# Determining Distribution using normal/chi-square

1. ### cse63146

452
1. The problem statement, all variables and given/known data

Suppose X,W,Y and Z are all independent. X & Y have a normal distribution. W has N(4,4) distribution while Z has a chi-square with 2 degrees of freedom.

a) What is the distribution of X2 + Y2 + Z?
b) What is the distribution of W - 4/(2|X|)
c) What is the distribution of (X2 + Y2)/Z

2. Relevant equations

3. The attempt at a solution

a) X~N(0,1) Y~N(0,1) Y2~X2(1) X2~X2(1)

Let S = X2 + Y2 + Z

therefore S~X2(4)

c) X~N(0,1) Y~N(0,1) Y2~X2(1) X2~X2(1)

Let S = (X2 + Y2)/Z

therefore S~X2(1)

b) Not sure what to do about the absolute value.

1,478
Your response to c isn't correct - you cannot say that because the numerator has 2 degrees of freedom, as does the denominator, they cancel.
You do know that the numerator and denominator are independent, and both have chi-square distributions, so ...

3. ### cse63146

452
Ah, they become an F-distribution

so S ~ F2,2

how would I handle the |X| in part b)?

4. ### Billy Bob

392
$$|X|=\sqrt{X^2}$$

452
Thank you.