Results 1 to 7 of 7

Math Help - Statistics: sampling distributions related to normal distribution

  1. #1
    Senior Member
    Joined
    Jan 2009
    Posts
    404

    Statistics: sampling distributions related to normal distribution


    iid=independent and identically distributed

    This is the model answer to a question which is unlikely to be wrong. I don't understand the part circled in red. How is that step justified? ~ is not the same as a equal sign, so can we still do that?
    In other words, my question is "If c is a constant and cW follows the t-distribution with parameter 12, i.e. cW~t(12), does this imply that W~1/c t(12)?"
    If so, WHY?
    I hope that someone can help me out. Thanks a lot!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by kingwinner View Post

    iid=independent and identically distributed

    This is the model answer to a question which is unlikely to be wrong. I don't understand the part circled in red. How is that step justified? ~ is not the same as a equal sign, so can we still do that?
    In other words, my question is "If c is a constant and cW follows the t-distribution with parameter 12, i.e. cW~t(12), does this imply that W~1/c t(12)?"
    If so, WHY?
    I hope that someone can help me out. Thanks a lot!
    Let X have a pdf f(x) and consider finding the pdf of U = \alpha X:

    The cdf of U is G(u) = \Pr(U < u) = \Pr(\alpha X < u) = \Pr\left( X < \frac{u}{\alpha}\right) = \int_{-\infty}^{u/\alpha} f(x) \, dx.

    Therefore the pdf of U is g(u) = \frac{dG}{du} = \frac{d}{du} \int_{-\infty}^{u/\alpha} f(x) \, dx = \frac{1}{\alpha} f\left( \frac{u}{\alpha}\right).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Jan 2009
    Posts
    404
    Quote Originally Posted by mr fantastic View Post
    Let X have a pdf f(x) and consider finding the pdf of U = \alpha X:

    The cdf of U is G(u) = \Pr(U < u) = \Pr(\alpha X < u) = \Pr\left( X < \frac{u}{\alpha}\right) = \int_{-\infty}^{u/\alpha} f(x) \, dx.

    Therefore the pdf of U is g(u) = \frac{dG}{du} = \frac{d}{du} \int_{-\infty}^{u/\alpha} f(x) \, dx = \frac{1}{\alpha} f\left( \frac{u}{\alpha}\right).
    Firstly, here we are assuming alpha>0, right?

    Secondly, the f(u/alpha) seems to make things complicated. I still can't see why "If c is a constant and cW follows the t-distribution with parameter 12, i.e. cW~t(12), then W~1/c t(12)". Could you explain a little bit further?

    Thank you!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by kingwinner View Post
    Firstly, here we are assuming alpha>0, right?

    Secondly, the f(u/alpha) seems to make things complicated. I still can't see why "If c is a constant and cW follows the t-distribution with parameter 12, i.e. cW~t(12), then W~1/c t(12)". Could you explain a little bit further?

    Thank you!
    On behalf of the mathematics (which is the cause of the trouble), I apologise that "the f(u/alpha) seems to make things complicated".

    My argument needs to be changed if \alpha < 0 because of the necessary reversal in the inequality as a result of dividing by \alpha < 0. I should have stated that, I just assumed \alpha > 0 because that was your specific circumstance. I will let you check what the end-result will be.

    I have given you a formula with a clear derivation of where it has come from. Apply the formula for your particular situation. In fact, feel free to use the derivation as a guide for deriving the formula for your own specific case.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Jan 2009
    Posts
    404
    But I haven't learnt about the density function of the t-distribution in my class. Do we need it in this case?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Jan 2009
    Posts
    404
    Now I am seeing another trouble...
    If X is a random variable, then I know what 1/c X means, but t(12) is a distribution, what is the meaning of saying some random variable follows the distribution 1/c t(12)?
    Last edited by kingwinner; January 16th 2009 at 09:19 AM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by kingwinner View Post
    Now I am seeing another trouble...
    If X is a random variable, then I know what 1/c X means, but t(12) is a distribution, what is the maning of saying some random variable follows the distribution 1/c t(12)?
    I have given you a formula. It tells you what the pdf of \alpha X is (for \alpha > 0) in terms of the pdf of X.

    In your case the pdf of X is the pdf for the t-distribution. The meaning of \frac{1}{\alpha} f\left( \frac{u}{\alpha}\right) is that it's the pdf of \alpha X. I really don't know what else to say.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Statistics Sampling Distribution
    Posted in the Statistics Forum
    Replies: 0
    Last Post: December 6th 2011, 12:17 PM
  2. Replies: 7
    Last Post: July 4th 2010, 08:46 AM
  3. Problem on normal/gaussian distribution sampling
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: July 3rd 2010, 09:28 PM
  4. Normal and Sampling Distribution
    Posted in the Statistics Forum
    Replies: 0
    Last Post: September 11th 2009, 07:01 PM
  5. Normal and Sampling Distribution #2
    Posted in the Statistics Forum
    Replies: 0
    Last Post: September 11th 2009, 07:01 PM

Search Tags


/mathhelpforum @mathhelpforum