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Math Help - Combinations of random variables

  1. #1
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    Cool Combinations of random variables

    Can anyone help me on the following Pls?

    The random variable X has p.d.f given by
    f(x) = x/2 0 x 2
    0 otherwise
    (a) Find the c.d.f of X, FX(x).
    The random variable Y = X2 and takes values over the range
    0 y 4.
    (b) Show that P(Y < y) = P(X y).
    (c) Hence show that the c.d.f of Y, FY(y) is given by

    FY(y) = 0, y < 0
    y/4, 0 y 4
    1, y > 4
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  2. #2
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    Quote Originally Posted by Sashikala View Post
    Can anyone help me on the following Pls?

    The random variable X has p.d.f given by
    f(x) = x/2 0 x 2
    0 otherwise
    (a) Find the c.d.f of X, FX(x).
    The random variable Y = X2 and takes values over the range
    0 y 4.
    (b) Show that P(Y < y) = P(X y).
    (c) Hence show that the c.d.f of Y, FY(y) is given by

    FY(y) = 0, y < 0
    y/4, 0 y 4
    1, y > 4
    Set up and do the required integrations. What have you tried? Where do you get stuck?
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  3. #3
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    Cool

    for the part(a) I got
    F(x)=0 , x< 0
    square of x/4, 0 x 2
    1, x>2

    Do we have to prove (b) with the
    help of probability distribution table?
    How to prove it using the p.d.f though
    both are equal?
    How to get at c.d.f of Y?
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  4. #4
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    Quote Originally Posted by Sashikala View Post
    for the part(a) I got
    F(x)=0 , x< 0
    square of x/4, 0 x 2
    1, x>2

    Do we have to prove (b) with the
    help of probability distribution table?
    How to prove it using the p.d.f though
    both are equal?
    How to get at c.d.f of Y?
    F_Y(y) = \Pr(Y < y) = \Pr(X^2 < y) = \Pr(-\sqrt{y} < X < \sqrt{y}) = \Pr(X < \sqrt{y})

    since \Pr(X < 0) = 0

    = F_X(\sqrt{y}) = \frac{y}{4}.

    f_Y(y) = \frac{dF_Y}{dy}.

    The details are left for you to fill in.
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  5. #5
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    Wink Combinations of random variables

    Yes. I understood.
    Thanks very much
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