1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: A question about statistics

  1. Feb 4, 2013 #1
    1. The problem statement, all variables and given/known data

    Let X1, X2, …, Xn be a random sample from a Poisson(λ) distribution. Let [itex]\bar{X}[/itex] be their sample mean and [itex]S^2[/itex] their sample variance.
    a) Show that [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{\sqrt{\bar{X}}}[/itex] and [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{S}[/itex] both have a standard normal limiting distribution.
    b) Find the limiting distribution of [itex]\sqrt{n}[\bar{X}-\lambda]^2[/itex]
    c) Find the limiting distribution of [itex]\sqrt{n}[\bar{X}^2-\lambda^2][/itex]

    2. Relevant equations

    3. The attempt at a solution

    a) For [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{S}[/itex], we know that [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{S}[/itex] = [itex](\frac{\sqrt{n}[\bar{X}-\lambda]}{\sigma})(\frac{\sigma}{S})[/itex]. Since
    [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{\sigma}[/itex] appraches N(0,1) in distribution by CLT, and since [itex](\frac{\sigma}{S})[/itex] appraches 1 in probability, the whole thing approaches N(0,1).

    [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{\sqrt{\bar{X}}}[/itex], we get the same result...since the mean is equal to the variance in a poisson distribution.

    b) I'm a little confused about this one...

    c) From a theorem in my textbook, I know that if [itex]\sqrt{n}(X_n - \theta) \rightarrow N(0,\sigma^2)[/itex] and if there is a differentiable function g(x) at theta where the derivative at theta is not zero...then [itex]\sqrt{n}(g(X_n)-g(\theta)) \rightarrow N(0,\sigma^2(g'(\theta))^2)[/itex].

    So I just need to use this theorem, right? And in this case g(x)=x^2.

    Do you think my answer for a), and c) are correct? Also, can you give me a hint for b)?

    Thanks in advance
    Last edited: Feb 4, 2013
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted