Results 1 to 7 of 7

Math Help - What is the distribution of ratios?

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    4

    What is the distribution of ratios?

    Hi,

    would appreciate the help.

    I am wondering if we have  X_1, X_2,X_3,X_4 with  Gamma(a, \theta_1) and another distribution with  Y_1, Y_2  Gamma(b,\theta_2) what would be the distribution of Y/X (Y and X are sample means) ?

    I know it would be Gamma, but not sure about the new alpha and Beta?

    What about other distributions?

    Thanks
    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 hitman View Post
    Hi,

    would appreciate the help.

    I am wondering if we have  X_1, X_2,X_3,X_4 with  Gamma(a, \theta_1) and another distribution with  Y_1, Y_2 Gamma(b,\theta_2) what would be the distribution of Y/X (Y and X are sample means) ?

    I know it would be Gamma, but not sure about the new alpha and Beta?

    What about other distributions?

    Thanks
    First get the distribution of \overline{Y} and \overline{X}. Are the X_i independent and the Y_i independent (using the moment generating function might then be one way of doing this).

    Then get the cdf of U = \frac{\overline{Y}}{\overline{X}}.

    Get the pdf from the cdf.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    Well, the sum if indep gamma's with the same beta is a gamma.
    The ratio of this random variable is similar to an F density.
    Thats a ratio of \chi^2 random variables divided by it's degrees of freedom.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Mar 2009
    Posts
    4
    Quote Originally Posted by matheagle View Post
    Well, the sum if indep gamma's with the same beta is a gamma.
    The ratio of this random variable is similar to an F density.
    Thats a ratio of \chi^2 random variables divided by it's degrees of freedom.
    Ah,, I am confused U_1/d_1/U_2/d_2 U must have \chi^2 distribution. But I don't understand how do X bar and Y bar have \chi^2 distribution?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by hitman View Post
    Ah,, I am confused U_1/d_1/U_2/d_2 U must have \chi^2 distribution. But I don't understand how do X bar and Y bar have \chi^2 distribution?

    Thanks
    They don't. I think you have misunderstood matheagles's reply.

    1. \overline{X} = \frac{X_1 + X_2 + X_3 + X_4}{4}.

    2. m_{X_i}(t) = \frac{1}{(1 - \theta t)^a} therefore m_{\frac{X_i}{4}}(t) = \frac{1}{\left(1 - \left(\frac{\theta}{4}\right) t\right)^a}.

    3. m_{\overline{X}}(t) = m_{\frac{X_1}{4}}(t) \cdot m_{\frac{X_2}{4}}(t) \cdot m_{\frac{X_3}{4}}(t) \cdot m_{\frac{X_4}{4}}(t)

    (assuming independence)


    = \left[ \frac{1}{\left(1 - \left(\frac{\theta}{4}\right) t\right)^a}\right]^4 = \frac{1}{ \left(1 - \left( \frac{\theta}{4}\right) t\right)^{4a}}


    which is recognised as the mgf of another gamma distribution.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Mar 2009
    Posts
    4
    I see,, I think I got the idea, I am going to play with it.

    Another question, about F-distribution, lets say  w \sim F(4,5) now we want to find a and b such that  P(W> a) = 0.05 = P(W<b)

    This question is not clear for me.  p(w>a) = 0.05 is easy. you calculate the degree of freedom, look into the table for 3,4 and 0.05 and you find a. However, I am not sure about the next part. is it like:

     p(w<b) = 1- p(w>b) \rightarrow   p(w>b) = 0.95

    and probably to find 0.95 I have to use statistic software
    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 hitman View Post
    I see,, I think I got the idea, I am going to play with it.

    Another question, about F-distribution, lets say  w \sim F(4,5) now we want to find a and b such that  P(W> a) = 0.05 = P(W<b)

    This question is not clear for me.  p(w>a) = 0.05 is easy. you calculate the degree of freedom, look into the table for 3,4 and 0.05 and you find a. However, I am not sure about the next part. is it like:

     p(w<b) = 1- p(w>b) \rightarrow p(w>b) = 0.95

    and probably to find 0.95 I have to use statistic software Mr F says: Yes, you do.
    ..
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. ratios and trig ratios
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: November 7th 2011, 09:45 AM
  2. Ratios
    Posted in the Algebra Forum
    Replies: 3
    Last Post: September 14th 2011, 04:40 PM
  3. Need help with ratios
    Posted in the Algebra Forum
    Replies: 1
    Last Post: August 4th 2011, 12:49 AM
  4. Ratios
    Posted in the Math Topics Forum
    Replies: 5
    Last Post: January 7th 2011, 09:11 AM
  5. multiplying ratios without knowing the ratios?
    Posted in the Statistics Forum
    Replies: 1
    Last Post: September 28th 2009, 07:59 AM

Search Tags


/mathhelpforum @mathhelpforum