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 $$P( Y < \frac{X}{2})$$

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

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

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]