Probability - Condition/Marginal density and Expectation

[cse63146]

Homework Statement

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

f(x,y) = 10x^2y if 0<x<y<1 0 othewise

a) Determine P( Y &lt; \frac{X}{2})

b) Determine P(x \leq 1/2 | Y &lt; X^2)

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

d) Determine E[XY^2]

e) Determine E[Y|X = x]

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)

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

b) P(A|B) = \frac{P(A \cap B)}{P(B)} \rightarrow \frac{P(X \leq 1/2 \cap Y &lt; X^2)}{P(Y &lt; X^2)}

c)

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

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

e)

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

f)

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

g) Not sure how to do this one.

 

[cse63146]

Any suggestions?

Got a type

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

f) Detetmine E[Y|X]