Question

Prove that sum of two independent chi square variates is a chi square variate

Answer 1

I want to prove that the sum of two independent chi-squared random variables is a chi-squared random variable.

I am supposed to only use the fact that if Q has a chi-squared distribution with parameter k then Q = Z21 + Z22 + ... + Z2k where each Zi is a standard normally distributed random variable and {Z1,...,Zk} is independent.

My attempt at a proof:

Let Q1 and Q2 be independent random variables with chi-squared distributions, with parameters a and b, respectively. Let {X1,...,Xa,Y1,...,Yb} be a set of independent random variables with standard normal distributions. Then we can write

Q1 = X21 + X22 + ... + X2a

Q2 = Y21 + Y22 + ... + Y2b , and Q1 and Q2 are independent because {X1,...,Xa,Y1,...,Yb} is independent.

so Q2 + Q2 = X21 + X22 + ... + X2a + Y21 + Y22 + ... + Y2b.

Since {X1,...,Xa,Y1,...,Yb} is independent, Q2 + Q2 is a chi-squared random variable with parameter a+b.

I don't think my proof is correct. I think the problem is that if we are given Q1 and Q2 that are independent, we can't just write them in terms of {X1,...,Xa,Y1,...,Yb}. But I am not really sure. Please tell me why my proof is incorrect (or maybe it is correct). Any help is appreciated.
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