Results 1 to 3 of 3

Thread: beta distribution probability density function proof

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    2

    Question beta distribution probability density function proof

    Hi
    I'm having bit of trouble proving this fact.
    beta distribution probability density function proof-beta-dist-density-.jpg
    can someone help me with the proof? or at least lead me thru it?

    Thanks

    -AC
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,002
    Thanks
    1
    Prove $\displaystyle B(p,q)=\int_{0}^{1}x^{p-1}(1-x)^{q-1}dx=\frac{{\Gamma}(p){\Gamma}(q)}{{\Gamma}(p+q)}$


    $\displaystyle B(p,q)=\frac{{\Gamma}(p){\Gamma}(q)}{{\Gamma}(p+q) }$....[1]

    Start with $\displaystyle {\Gamma}(p)=\int_{0}^{\infty}t^{p-1}e^{-t}dt$

    Now, put $\displaystyle t=y^{2}$, and we get:

    $\displaystyle {\Gamma}(p)=2\int_{0}^{\infty}y^{2p-1}e^{-y^{2}}dy$....[2]

    Similarly, (the dummy integration variable can be any letter):

    $\displaystyle {\Gamma}(p)=2\int_{0}^{\infty}x^{2q-1}e^{-x^{2}}dx$

    Next, we multiply these 2 equations and change to polar coordinates:

    $\displaystyle {\Gamma}(p){\Gamma}(q)=4\int_{0}^{\infty}\int_{0}^ {\infty}x^{2q-1}y^{2p-1}e^{-(x^{2}+y^{2})}dxdy$

    $\displaystyle =4\int_{0}^{\infty}\int_{0}^{\frac{\pi}{2}}(rcos{\ theta})^{2q-1}(rsin{\theta})^{2p-1}e^{-r^{2}}rdrd{\theta}$

    $\displaystyle =4\int_{0}^{\infty}r^{2p+2q-1}e^{-r^{2}}dr\int_{0}^{\frac{\pi}{2}}(cos{\theta})^{2q-1}(sin{\theta})^{2p-1}d{\theta}$....[3]

    The r integral in [3] is $\displaystyle \frac{1}{2}{\Gamma}(p+q)$ by [2].

    The $\displaystyle {\theta}$ in [3] is $\displaystyle \frac{1}{2}B(p,q)$ by

    $\displaystyle B(p,q)=2\int_{0}^{\frac{\pi}{2}}(sin{\theta})^{2p-1}(cos{\theta})^{2q-1}d{\theta}$

    Then, $\displaystyle {\Gamma}(p){\Gamma}(q)=4\cdot \frac{1}{2}{\Gamma}(p+q)\cdot \frac{1}{2}B(p,q)$ and [1] follows.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2008
    Posts
    2
    Wow! that was quick!
    Thanks for the help!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. probability distribution and density function
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: Jan 9th 2011, 12:34 PM
  2. Probability Density of Normal Distribution
    Posted in the Statistics Forum
    Replies: 1
    Last Post: Nov 20th 2009, 12:50 AM
  3. Beta function proof
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Jun 14th 2009, 03:03 AM
  4. Proof for traditional mean of Pert formula (Beta distribution)
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Apr 27th 2009, 04:05 PM
  5. Beta Distribution Proof
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: Jan 29th 2009, 01:50 PM

Search tags for this page

Search Tags


/mathhelpforum @mathhelpforum