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Math Help - Stats review

  1. #1
    Junior Member plm2e's Avatar
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    Stats review

    hello, I am working on a review sheet to get ready for a stats test. I am not finding any examples in my text book to help me. I have a few questions and I am looking for an explanation, not just the answer. Please help me out if you can.


    1. Let X1, X2, ... ,Xn be independent random variables, each having a uniform distribution over (0,1). Let M = maximum (X1, X2, ... ,Xn). Show that the distribution function of M, FM(.) is given by: FM(x) = x^n , 0< x < 1.


    2. The joint density of X and Y is given by:

    f(x,y) = [e^(-x/y)e^(-y)]/y , 0 < x < , 0 < y <

    Show that E[XlY=y] = y


    3. If X and Y are both discrete, show that ∑x Pxly (xly)= 1 for all such that PY(y) > 0.


    4. Show that
    a. Cov (X,Y) = Cov (X,E[YlX])
    b. suppose, that, for constants a and b, E[YlX] = a + bX
    show that b= Cov(X,Y)/Var(X)
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  2. #2
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    Quote Originally Posted by plm2e View Post
    hello, I am working on a review sheet to get ready for a stats test. I am not finding any examples in my text book to help me. I have a few questions and I am looking for an explanation, not just the answer. Please help me out if you can.


    1. Let X1, X2, ... ,Xn be independent random variables, each having a uniform distribution over (0,1). Let M = maximum (X1, X2, ... ,Xn). Show that the distribution function of M, FM(.) is given by: FM(x) = x^n , 0< x < 1.

    [snip]
    F_M (x) = \Pr(M \leq x) = \Pr(X_1 \leq x, \, X_2 \leq x, \, .... \, X_n \leq x) = \Pr(X_1 \leq x) \cdot \Pr(X_2 \leq x) \cdot .... \Pr(X_n \leq x) = \left[ G(x) \right]^n

    (since the X's are i.i.d) where G(x) is the cdf of X.

    And you should know that G(x) = \int_{0}^{x} 1 \, du = x for 0 \leq x \leq 1.
    Last edited by mr fantastic; October 15th 2008 at 07:29 PM.
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  3. #3
    Junior Member plm2e's Avatar
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    thanks for the help on prob 1.
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  4. #4
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    Quote Originally Posted by plm2e View Post
    [snip]

    2. The joint density of X and Y is given by:

    f(x,y) = [e^(-x/y)e^(-y)]/y , 0 < x < , 0 < y <

    Show that E[XlY=y] = y

    [snip]
    From the definition: E(X \, | \, Y = y) = \int_{0}^{+\infty} x \, f(x \, | \, Y = y) \, dx.

    From the definition: f(x \, | \, Y = y) = \frac{f(x, y)}{f_Y (y)}.

    From the definition: f_Y(y) = \int_{0}^{+\infty} f(x, y) \, dx = \int_{0}^{+\infty} \frac{e^{-x/y} \, e^{-y}}{y} \, dx = \frac{e^{-y}}{y} \int_{0}^{+\infty} e^{-x/y} \, dx = e^{-y}.
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