Finding Expected Value with Joint Density: (3X+4Y)/(X+2Y)

[nikki92]

Homework Statement

(X,Y) has joint density f_X,Y ( (x,y) = (3/16)(x+2y) for 0<y<x , 0< x<2

Evaluate E(Z) where Z = (3X+4Y)/(X+2Y)

Homework Equations

The Attempt at a Solution

Getting the marginal densities
f_X (x) =(3/8)x^2 for 0<x<2

f_Y (y) = (3/2)+(3/4)y for 0<y<2

Would I find the new distribution of 3X , 4Y, and 2Y then do the ratio distribution to solve for the distribution of Z? If this is correct, is there a shorter way?

 

[hedipaldi]

E[3X+4Y/X+2Y]=3E[X/X+2Y]+4E[Y/X+2Y]

now use fx/z=f(x,z)/f(z) and integration over suitable rigions in the xy plane.
fy/z=f(y,z)/f(z)
recall that E[x/z]=integral(xfx/z)

 

[Ray Vickson]

Homework Statement

(X,Y) has joint density f_X,Y ( (x,y) = (3/16)(x+2y) for 0<y<x , 0< x<2

Evaluate E(Z) where Z = (3X+4Y)/(X+2Y)

Homework Equations

The Attempt at a Solution

Getting the marginal densities
f_X (x) =(3/8)x^2 for 0<x<2

f_Y (y) = (3/2)+(3/4)y for 0<y<2

Would I find the new distribution of 3X , 4Y, and 2Y then do the ratio distribution to solve for the distribution of Z? If this is correct, is there a shorter way?

You don't need to find marginal distributions (which would be of no help anyway) and there is no need to find the distribution of Z (although doing so would not be harmful). Just use the so-called "theorem (or law) of the unconscious statistician"; use Google if you have never heard of this.

RGV

 

[hedipaldi]

I tought the symbol / denotes conditional variable,but maybe you ment fraction.

 

[nikki92]

Oh yeah! Using the marginals would lose information.

So E[Z]=integral 0 to 2 integral 0 to x (3x+4y)/(x+2y) (3/16)(x+2y) dy dx = integral 0 to 2 (3/16)(3x+4y) dx = 5/2 Am I using the Unconscious Statistician theorem correctly? Thanks for all the help! <3

 

[Ray Vickson]

Oh yeah! Using the marginals would lose information.

So E[Z]=integral 0 to 2 integral 0 to x (3x+4y/x+2y) (3/16)(x+2y) dy dx = integral 0 to 2 3x^2(3x+1)/8 dx = 5/2 Am I using the Unconscious Statistician theorem correctly? Thanks for all the help! <3

Well, you should write (3x+4y)/(x+2y), not (3x+4y/x+ 2y) --- because this last means
$$3x + \frac{4y}{x} + 2y.$$
Also, when I do the y-integral I do not get 3x^2(3x+1)/8. This last integrates to 11/2 as x goes from 0 to 2, so I don't know how you get 5/2. (The 5/2 is correct; your computation is not.)

RGV

 

[nikki92]

Well, you should write (3x+4y)/(x+2y), not (3x+4y/x+ 2y) --- because this last means
$$3x + \frac{4y}{x} + 2y.$$
Also, when I do the y-integral I do not get 3x^2(3x+1)/8. This last integrates to 11/2 as x goes from 0 to 2, so I don't know how you get 5/2. (The 5/2 is correct; your computation is not.)

RGV

Thanks again! I typed the function incorrectly, but wrote it down correctly.