- #1
Askhwhelp
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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?
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?