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Math Help - Transforming Random Variables

  1. #1
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    Transforming Random Variables

    Let X1 and X2 be independent chi-square random variables with r1 and r2 degrees of freedom, respectively. Show that:

    a) U = X1/(X2 + X1) has a beta distribution with alpha = r1/2 and beta = r2/2

    b) V = X2/(X2 + X1) has a beta distribution with alpha =r2/2 and B = r1/2


    How do you even start this? I need a lot of help with this one.
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  2. #2
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    Quote Originally Posted by wolverine21 View Post
    Let X1 and X2 be independent chi-square random variables with r1 and r2 degrees of freedom, respectively. Show that:

    a) U = X1/(X2 + X1) has a beta distribution with alpha = r1/2 and beta = r2/2

    b) V = X2/(X2 + X1) has a beta distribution with alpha =r2/2 and B = r1/2


    How do you even start this? I need a lot of help with this one.
    I'll start a) (Obviously b) starts in a similar way):

    The cdf of U is

    F(u) = \Pr(U \leq u) = \Pr\left( \frac{X_1}{X_2 + X_1} \leq u\right) = \Pr(X_1 \leq uX_1 + uX_2)

    (the inequality remains in the same direction since X_2 + X_1 \geq 0)

    = \Pr (X_1 - u X_1 \leq uX_2) = \Pr(uX_2 \geq (1 - u) X_1) = \Pr\left( X_2 \geq \frac{1 - u}{u} \, X_1\right).

    Case 1: 0 < u < 1: F(u) = \int_{x_1 = 0}^{+\infty} \int_{x_2 = \frac{1 - u}{u} \, x_1}^{+\infty} f(x_1) \cdot g(x_2) \, dx_2 \, dx_1

    since the joint pdf of X_1 and X_2 is f(x_1) \cdot g(x_2) (because X_1 and X_2 are independent).

    The rest is left for you to do. Note that the pdf of U is given by h(u) = \frac{dF}{du}.


    Case 2: u > 1: F(u) = \int_{x_1 = 0}^{+\infty} \int_{x_2 = 0}^{+\infty} f(x_1) \cdot g(x_2) \, dx_2 \, dx_1 = \int_{x_1 = 0}^{+\infty}  f(x_1) \, dx_1 \, \int_{x_2 = 0}^{+\infty} g(x_2) \, dx_2 = 1

    \Rightarrow h(u) = \frac{dF}{du} = 0.
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