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Math Help - Mathematical Expectation

  1. #1
    Yan
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    1. If the joint probability density of X and Y is given by
    f(x,y) = (x+y)/3 for 0<x<1, 0<y<2 and 0 elsewhere.

    • Find the variance of W=3(X) +4(Y) - 5.


    2.If X1, X2, and X3 are independent and have the means 4,9, and 3 and the variances 3,7, and 5, find the mean and the variance of
    (a) Y=2(X1) - 3(X2) + 4(X3);
    (b) Z=(X1) + 2(X2) - (X3);
    (c) find cov(Y,Z).
    Last edited by mr fantastic; February 1st 2009 at 05:04 PM. Reason: Merged posts related to same question, minor editing for flow.
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  2. #2
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    Quote Originally Posted by Yan View Post
    1. If the joint probability density of X and Y is given by

    f(x,y) = (x+y)/3 for 0<x<1, 0<y<2 and 0 elsewhere.


    Find the variance of W=3(X) +4(Y) - 5.
    Find E(2X-Y)
    (b) Express var(X+Y), var (X-Y), and cov (X+Y, X-Y) in terms of the variances and covariance of X and Y.



    2.If X1, X2, and X3 are independent and have the means 4,9, and 3 and the variances 3,7, and 5, find the mean and the variance of

    (a) Y=2(X1) - 3(X2) + 4(X3);

    (b) Z=(X1) + 2(X2) - (X3);

    (c) find cov(Y,Z).
    Most of these questions are done by applying the basic definitions. Where are you stuck? What have you tried?
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  3. #3
    Yan
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    Quote Originally Posted by mr fantastic View Post
    Most of these questions are done by applying the basic definitions. Where are you stuck? What have you tried?
    the first question. is it using (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2), (3,0), (3,1), (3,2) and the another question, I really don't know how to do it.


    can you help me to do the second question? I don't know how to start it...
    Last edited by mr fantastic; January 29th 2009 at 08:46 PM.
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    Lord of certain Rings
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    Quote Originally Posted by Yan View Post
    can you help me to do the second question? I don't know how to start it...
    Here are things you need to know to solve second problem:

    [ \mathbb{E}(Y) stands for expectation of Y or the mean of Y]

    1) \mathbb{E}(X_1 + X_2 + X_3 + \cdots + X_n) =  \mathbb{E}(X_1) + \mathbb{E}(X_2) + \mathbb{E}(X_3) + \cdots + \mathbb{E}(X_n)

    2)If {X_i} s are all independent, then \text{Var}(X_1 + X_2 + X_3 + \cdots + X_n) =  \text{Var}(X_1) + \text{Var}(X_2) + \text{Var}(X_3) + \cdots + \text{Var}(X_n)

    3) \text{Var}(aX) = a^2 \text{Var}(X)

    4) Cov(Y,Z) = \mathbb{E}(YZ) -  \mathbb{E}(Y)\mathbb{E}(Z)
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  5. #5
    Yan
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    Quote Originally Posted by mr fantastic View Post
    Most of these questions are done by applying the basic definitions. Where are you stuck? What have you tried?
    then how to do this one?
    1. If the joint probability density of X and Y is given by

    f(x,y) = (x+y)/3 for 0<x<1, 0<y<2 and 0 elsewhere.


    Find the variance of W=3(X) +4(Y) - 5.
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  6. #6
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    Quote Originally Posted by Yan View Post
    then how to do this one?
    1. If the joint probability density of X and Y is given by

    f(x,y) = (x+y)/3 for 0<x<1, 0<y<2 and 0 elsewhere.


    Find the variance of W=3(X) +4(Y) - 5.
    You're given the joint pdf.

    The marginal pdf's are:

    f_X(x) = \int_0^2 \frac{x + y}{3} \, dy = \frac{2}{3} (x + 1).

    f_Y(y) = \int_0^1 \frac{x + y}{3} \, dx = \frac{1}{6} (1 + 2y).

    To get Var(W) = Var(3X + 4Y - 5), apply the usual formula to get this in terms of Var(X), Var(Y) and Cov(X, Y).

    Calculate Var(X), Var(Y) and Cov(X, Y) using the joint pdf and marginal pdf's.
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