Determining Distribution using normal/chi-square

[cse63146]

Homework Statement

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

Homework Equations

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.

 

[statdad]

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 ...

 

[cse63146]

Ah, they become an F-distribution

so S ~ F_2,2

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

 

[Billy Bob]

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

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

 

[cse63146]

Thank you.