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Math Help - Joint PDF with two random variables X and Y

  1. #1
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    Joint PDF with two random variables X and Y

    Hi,
    I have: PDF=f(x,y)=x+y with 0<=x<1 and 0<=y<=1
    I need to calculate: E(W) and Variance of (W) where: W=2X-Y

    also, they are asking for: prob (X<0.5 intersection with Y>0.5)

    Thanks
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  2. #2
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    Re: Joint PDF with two random variables X and Y

    Hey zaher11.

    Hint: E[W] = E[g(X,Y)] = Integral ([0,1]x[0,1])*g(X,Y)*f(x,y)dydx.
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  3. #3
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    Re: Joint PDF with two random variables X and Y

    thank you, the result =7/12

    do you know how to do: also, they are asking for: prob (X<0.5 intersection with Y>0.5) for: F(x,y)=x+y???
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  4. #4
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    Re: Joint PDF with two random variables X and Y

    For this one you need to get the region of integration.

    Since both are between 0 and 1 and both are independent, then the limits will be X = 0 to 0.5 and Y = 0.5 to 1.

    The rest is just evaluating the integral with those limits.
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  5. #5
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    Re: Joint PDF with two random variables X and Y

    thank you, but the probability will equal only to the double integral with limits like you mentioned, or it should be equal to the double integral you mentioned and divided by another double integral of P(Y>0.5)=double integral of (x+y) with x from 0 to 1 and y from 0.5 to 1; becasue we have intersection of two probabilites. Can you think about it please and tell me?
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  6. #6
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    Re: Joint PDF with two random variables X and Y

    What you should do for your own confirmation, is to draw the region that is being integrated on paper and convince yourself from that what the actual limits of integration should be.

    If this shows a different answer from mine, then please just point out with some graphical evidence where the inconsistency is.
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  7. #7
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    Re: Joint PDF with two random variables X and Y

    hi, you are right thnx.
    For the same, f(x,y) =x+y, I need to calculate:
    1) Pr(X<Y) so my double integral goes from X to 1 for Y, and from 0 to 1 for X, is this right?
    2) Pr((X+Y)<1) so my double integral goes from 0 to (Y-1) for X, and from 0 to 1 for Y, is this right?
    Thanks in advance
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  8. #8
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    Re: Joint PDF with two random variables X and Y

    For the first one, I think it should be Y from 0 to 1 and X from 0 to Y. Did you make a change of order of integration for your limits?
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  9. #9
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    Re: Joint PDF with two random variables X and Y

    Hi,
    for first one, Pr(X<Y)=double integral (x+y)dydx with first integral i.e y varies from X to 1 and X varies from 0 to 1
    is that right?
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  10. #10
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    Re: Joint PDF with two random variables X and Y

    hi, both of them gave me same result =1/2
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  11. #11
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    Re: Joint PDF with two random variables X and Y

    Joint PDF with two random variables X and Y-question.jpg
    Hi, in the attached question ,they are asking me to select what type of distribution I suggest to use for each part of the question? i.e: whether, normal distribution, poisson, uniform, bernoulli ...etc

    Please have a look and tell me what you think is the best distribution??

    Thanks
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