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Expected value of X and Y, E[XY] for uniform random variables

  1. Nov 10, 2014 #1
    1. The problem statement, all variables and given/known data
    If ##X\sim\mathcal{U}(-1,1)## and ##Y = X^2##, is it possible to determine to ##cov(X, Y)##?

    2. Relevant equations
    \begin{align}
    f_x &=
    \begin{cases}
    1/2, & -1<x<1\\
    0, & \text{otherwise}
    \end{cases}\\
    f_y &=
    \begin{cases}
    1/\sqrt{y}, & 0<x<1\\
    0, & \text{otherwise}
    \end{cases}
    \end{align}
    3. The attempt at a solution
    $$
    cov(X,Y) = E[XY] - E[X]E[Y] = E[XY] - 0\cdot 1/2 = E[XY]
    $$
    Now
    $$
    E[XY] = \int_0^1\int_{-1}^1g(X, Y)f_{x,y}(x,y)dxdy
    $$
    From the information that I have, can I determine ##E[XY]##?
     
  2. jcsd
  3. Nov 10, 2014 #2

    Orodruin

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    I suggest using the fact that ##Y = X^2##...
     
  4. Nov 10, 2014 #3
    How?
     
  5. Nov 10, 2014 #4

    Orodruin

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    What do you get if you replace ##Y## everywhere by ##X^2##?
     
  6. Nov 10, 2014 #5

    Ray Vickson

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    For any observed value ##X = x##, the observed (or, rather, computed) value of ##Y## is ##y = x^2##. It not just that ##Y## and ##X^2## have the same distribution; much more than that is true: ##Y## IS ##X^2##.
     
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