Sums and products of random variables

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Can anyone help me with the below question?

for each of the following pairs of random variables X,Y, indicate
a. whether X and Y are dependent or independent
b. whether X and Y are positively correlated, negatively correlate or uncorrelated

i. X and Y are uniformly distributed on the disk {(x,y) in R^2: 0<=x^2+y^2<=1}
since this is a circle with R=1 the area is pi. Since x and y are uniform then fxy(xy) is 1/area = 1/pi. In order to see if x and y are independent i need to computer marginal densities and multiply to see if i get fxy(xy). This is where I am stuck. I know that to get fx i need to integrate fxy(xy) with respect to dy and for fy integrate with respect to dx. But what am i integrating? if fx = integral from -1 to 1 of 1/pi dy? and fy = to integral from -1 to 1 1/pi dx? For correlation i first need COV(XY) which equals = E(XY)-E(X)E(Y). To get E(XY) i integrate xy*fxy(xy) so the double integral from -1 to 1 of xy/pi? Is this correct? Now, since x and y are uniform then E(X) and E(Y) are just the interval over two so since each is between -1 and 1 we get 2/2=1=E(X)=E(Y). The VAR(X) and VAR(Y) is 2^/12, the radical of which gives standard deviation and from these value we can compute rho.

ii. X and Y are uniformly distributed on the parallelogram {(x,y) in R^2: x-1<=y<=x+1, -1<=x<=1}. the area of the parallelogram is 4, so fxy(xy) is 1/4. to get fx i integrate 1/4 from x-1 to x+1 with respect to dy which yields 1/2. To get fy i integrate 1/4 from -1 to 1 with respect to dx which also yields 1/2. since fx*fy=1/4=fxy(xy) then x and y are independent and thus uncorrelated. correct?

X and Y are uniformly distributed on the diamond {(x,y) in R^2: |x|+|y|<=1, |x|<=1}. i am not sure about this one. what will the graph of this one look like? is it a diamond on -1<=x<=1 and -1<=y<=1?
 
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Answers and Replies

  • #2
Stephen Tashi
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. But what am i integrating? if fx = integral from -1 to 1 of 1/pi dy? and fy = to integral from -1 to 1 1/pi dx?

What is your background in calculus? Have you done multiple integration problems where the limits of integration on one variable were stated as a function of the other variable?
 

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