Question

The joint distribution of two continuous random variables $X$ and $Y$ are given by: $f_{X,Y}(x,y) = Cxy$, for $0\leq x\leq y\leq 1$, and $0$ elsewhere. 1. Find $C$ to make $f_{X,Y}(x,y)$ a valid probability density function. Enter the numerical value of $C$ here: 2. What should be the correct PDF for $f_X(x)$: A. $f_X(x) = 2x$ for $0\leq x\leq 1$, and $0$ elsewhere. B. $f_X(x) = 3x^2$ for $0\leq x\leq 1$, and $0$ elsewhere. C. $f_X(x) = 4x(1-x^2)$ for $0\leq x\leq 1$, and $0$ elsewhere. D. $f_X(x) = \frac{3}{2}(1-x^2)$ for $0\leq x\leq y\leq 1$, and $0$ elsewhere. Enter your answer for subquestion 2. here (only A, B, C, or D is accepted) 3. What should be the correct PDF for $f_Y(y)$: A. $f_Y(y) = 2y$ for $0\leq y\leq 1$, and $0$ elsewhere. B. $f_Y(y) = 3y^2$ for $0\leq y\leq 1$, and $0$ elsewhere. C. $f_Y(y) = 4y(1-y^2)$ for $0\leq y\leq 1$, and $0$ elsewhere. D. $f_Y(y) = 4y^3$ for $0\leq y\leq 1$, and $0$ elsewhere. Enter your answer for subquestion 3. here (only A, B, C, or D is accepted) 4. Are $X$ and $Y$ independent? If YES, enter Y, if NO, enter N. Enter your answer here for subquestion 4. here (only Y or N is accepted) 5. Find the covariance of $X$ and $Y$. Enter your answer here for subquestion 5. here (round up to 4 decimal points)

Answer 1

The joint pdf of X and Y is

1) To make f(x,y) a valid pdf, it has to integrate to 1 over the ranges of X and Y

Ans C=8

the joint pdf of X,Y is

2. the marginal pdf of X is

Formally the pdf of X is

ans: C.

3) The marginal pdf of Y is

Formally the pdf of Y is

ans: D

4) X and Y are independent if the product of marginal pdfs of X and Y is equal to the joint pdf of X and Y

Here, the product is

Since the product of marginal pdfs of X and Y is not equal to the joint pdf of X and Y

X and Y are not independent

ans: N

5) The covariance of X,y is

First the expectations

The expectation of X is

the expectation of Y is

the expectation of XY is

The covariance of X and Y is

ans: 0.0178