Chi-square distribution Verification

[Askhwhelp]

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?

 

[mighty2000]

Your approach is correct in terms of determining whether the given data fits the definition of a chi-square distribution. However, it is important to note that the question is asking whether Y itself has a chi-square distribution. Therefore, the approach should be to determine if Y fits the definition of a chi-square distribution, rather than calculating the expected values.

To determine if Y has a chi-square distribution, we need to check if it is a gamma-distributed random variable with parameters α = ν/2 and β = 2. This can be done by checking if the probability density function of Y matches the gamma distribution with these parameters.

Another approach could be to use statistical tests such as the chi-square goodness of fit test to determine if the data fits a chi-square distribution. This test compares the observed data to the expected data under the assumption of a chi-square distribution and provides a p-value to determine the significance of the fit.

In summary, to determine if Y has a chi-square distribution, we need to check if it meets the definition of a gamma-distributed random variable with parameters α = ν/2 and β = 2 or use statistical tests to assess the fit of the data to a chi-square distribution.