1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Chi-square distribution Verification

  1. Dec 10, 2013 #1
    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?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?