1. ## Joint Density problem....

Question 5.
The joint density of two random variables X and Y is given by
f
(x, y) = (x^2)y
whenever
0 < x < 1 and 0 < y < 4, and f (x, y) = 0 otherwise.
Are X and Y independent? Justify

EX, and find the probability that X +Y is less than 3.

Well the problem is, you think that this is surely independent since it factors, but my teacher said something about how to be sure, you need to find the marginal densities and they need to multiply to give f(x,y), but when I compute the marginal densities, they don't which would mean they are dependent, but I could be doing something wrong, so any ideas which it is?
I can get EX, but for P(X+Y<3), can someone help set up the integral for me. I can compute it myself, but setting it up is the trouble.

2. Originally Posted by zhupolongjoe
Question 5.
The joint density of two random variables X and Y is given by

f (x, y) = (x^2)y
whenever 0 < x < 1 and 0 < y < 4, and f (x, y) = 0 otherwise.
Are X and Y independent? Justify
your answer. Compute EX, and find the probability that X +Y is less than 3.

Well the problem is, you think that this is surely independent since it factors, but my teacher said something about how to be sure, you need to find the marginal densities and they need to multiply to give f(x,y), but when I compute the marginal densities, they don't which would mean they are dependent, but I could be doing something wrong, so any ideas which it is?
I can get EX, but for P(X+Y<3), can someone help set up the integral for me. I can compute it myself, but setting it up is the trouble.

A posted, you do not have a valid pdf: $\int_{x = 0}^{1} \int_{y = 0}^4 x^2 y \, dy \, dx \neq 1$. So there's no point attempting these questions.

3. Originally Posted by zhupolongjoe
Question 5.

The joint density of two random variables X and Y is given by

f (x, y) = (x^2)y
whenever 0 < x < 1 and 0 < y < 4, and f (x, y) = 0 otherwise.
Are X and Y independent? Justify
your answer. Compute EX, and find the probability that X +Y is less than 3.

Well the problem is, you think that this is surely independent since it factors, but my teacher said something about how to be sure, you need to find the marginal densities and they need to multiply to give f(x,y), but when I compute the marginal densities, they don't which would mean they are dependent, but I could be doing something wrong, so any ideas which it is?
I can get EX, but for P(X+Y<3), can someone help set up the integral for me. I can compute it myself, but setting it up is the trouble.

I don't know much probability but I can help you set up the integral.

so we want to evaluate the integral for the values

$1> x > 0, y> 0,x+y< 3$
If we sketch a graph in the plane it will help us set it up

So we will integrate from the line $y=-x+3$ to 0 in the y direction.

$\int_{0}^{1} \int_{0}^{-x+3}x^2ydydx$

4. Hmmm...I copied it exactly from the review sheet...I don't know why my teacher would put such a trick...