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Probability - Condition/Marginal density and Expectation

  1. Mar 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Let X and Y be contnious random variables with joint probability density function -

    [tex]f(x,y) = 10x^2y[/tex] if 0<x<y<1 0 othewise

    a) Determine [tex]P( Y < \frac{X}{2})[/tex]

    b) Determine [tex]P(x \leq 1/2 | Y < X^2)[/tex]

    c) Determine the marginal density functions of X and Y, respectively

    d) Determine [tex]E[XY^2][/tex]

    e) Determine [tex]E[Y|X = x][/tex]

    g) Obtaine the probability density function of E[Y|X]

    2. Relevant equations



    3. The attempt at a solution

    Did I set up the a - f correctly?

    a)

    [tex]\int^1_0\int^{X/2}_0 10x^2y dy dx[/tex]

    b) [tex]P(A|B) = \frac{P(A \cap B)}{P(B)} \rightarrow \frac{P(X \leq 1/2 \cap Y < X^2)}{P(Y < X^2)}[/tex]

    c)

    [tex]F_Y (y) = \int^1_y 10x^2y dx[/tex] [tex]F_X (x) = \int^x_0 10x^2y dy[/tex]

    d)
    [tex]E[XY^2] = \int^1_0\int^x_0 xy^2 10x^2y dy dx[/tex]

    e)

    [tex]F_{Y|X} (Y|X) = \frac{f(x,y)}{F_X (x)}[/tex]

    f)

    [tex]E[Y|X] = \int^y_0 y F_{Y|X} (Y|X) dy [/tex]

    g) Not sure how to do this one.
     
  2. jcsd
  3. Mar 20, 2009 #2
    Any suggestions?

    Got a type

    e) Determine conditional density function of Y given X = x.

    f) Detetmine E[Y|X]
     
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