# Chi-square distribution Verification

1. Dec 10, 2013

By Definition, Let ν be a positive integer. A random variable Y is said to have a chi-square distribution with ν degrees of freedom if and only if Y is a gamma-distributed random variable with parameters α = ν/2 and β = 2.

By Thm, If Y is a chi-square random variable with ν degrees of freedom, then
μ = E(Y) = ν and $σ^2$ = V(Y) = 2ν.

The question to test whether Y, $E(Y) = 10$ and $E[(1+Y)^2] = 36$ has a chi-square distribution?

(1) Whether $E(Y) = 10$, $E[(1+Y)^2] = 36$ have a chi-square distribution? Explain why, why not, or cannot be determined.

My approach was

$E(Y^2) = V(Y)+[E(Y)]^2 = 2*10 + 10^2 = 120$E(Y) = 10, E[(1+Y)^2] = E[1+2Y+Y^2] = E(1) + 2E(Y) + E(Y^2) = 1 + 2*10 + 120 = 141 $Therefore, this is a not chi-square distribution. (2)Whether$E(Y) = 10$,$E[(1+Y)^2] = 51$have a chi-square distribution? Explain why, why not, or cannot be determined. For the same reasoning above, this is a not chi-square distribution. However, after taking a closer look of the question: the question is asking whether Y,$E(Y)=10$and$E[(1+Y)2]=36\$, have a chi-square distribution...however, by definition, a random variable Y is said to have a chi-square distribution with ν degrees of freedom if and only if Y is a gamma-distributed random variable with parameters α = ν/2 and β = 2. ...

The question we need to answer does not mention anything about with degrees of freedom...so is my argument with using parameters α = ν/2 and β = 2 valid?

If valid, am my approach right? If not, what is the right approach? If so, is there any other ways to show this?