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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.
(1) Whether E(Y) = 10, E[(1+Y)^2] = 36 have a chi-square distribution?
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?
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.
Am my reasoning right? If not, what is the right approach? If so, is there any other ways to show this?
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.
(1) Whether E(Y) = 10, E[(1+Y)^2] = 36 have a chi-square distribution?
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?
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.
Am my reasoning right? If not, what is the right approach? If so, is there any other ways to show this?