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Math Help - Help with a density function

  1. #1
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    Help with a density function

    Question:
    Suppose that random variables X and Y have a joint density function given by


    f_{X,Y} (x,y)=\left\{\begin{array}{cc}x+y,&\mbox{ for }<br />
0 \leq x\leq 1, 0\leq y\leq 1\\0, & \mbox{ otherwise } \end{array}\right.<br />

    a) \mbox {Find the density functions of X and Y,} i.e., f_X(x) \mbox {and} f_Y(y)
    b)\mbox {Find E[X] and }Var[Y]

    What I have done so far:
    Frankly I have been staring at this problem for a while. I'm not sure where to start. I cannot apply the following definition for joint probability functions
    f_{X,Y}(x,y)=f_X(x)\bullet f_Y(y) as the probability function is not independent. Anyone care to point me in the right direction? Once I could get it figured out, I can solve for b.

    PS. Please don't post any solutions as it defeats the whole purpose of asking for help, I won't learn anything that way.
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  2. #2
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    Something just hit me, in order to find f(x) do I compute the density function by integrating in respect to dy?
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  3. #3
    MHF Contributor harish21's Avatar
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    f_X(x)=\displaystyle \int_y f_{X,Y}(x,y) dy

    f_{Y}(y)=\displaystyle \int_x f_{X,Y}(x,y) dx

    After you have the marginal distributions, you can easily find the mean and variances of each.
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  4. #4
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    Quote Originally Posted by harish21 View Post
    f_X(x)=\displaystyle \int_y f_{X,Y}(x,y) dy

    f_{Y}(y)=\displaystyle \int_x f_{X,Y}(x,y) dx

    After you have the marginal distributions, you can easily find the mean and variances of each.
    Thanks a lot! It confirms what I said!
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