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**1. 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]

**2. Homework 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.