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cse63146
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Homework Statement
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]
Homework Equations
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.