Probablity Question - Joint PDF Expectation/Variance

[MCooltA]

Homework Statement

I have been given a joint PDF for X and Y, with ranges Y>X≥0.

I need to find the E(x) and E(y).

Homework Equations

I know E(x) = ∫(x)*(f(x,y) dx and E(y) = ∫(y)*(f(x,y)) dy

The Attempt at a Solution

For ∫x*f(x,y) dx, i used the limits = x to ∞

For ∫y*f(x,y) dy, i used the limits = 0 to y

Is this correct?

 

[Ray Vickson]

Homework Statement

I have been given a joint PDF for X and Y, with ranges Y>X≥0.

I need to find the E(x) and E(y).

Homework Equations

I know E(x) = ∫(x)*(f(x,y) dx and E(y) = ∫(y)*(f(x,y)) dy

The Attempt at a Solution

For ∫x*f(x,y) dx, i used the limits = x to ∞

For ∫y*f(x,y) dy, i used the limits = 0 to y

Is this correct?

These are both wrong. EX = int x*f(x,y) dx dy, etc.

RGV

 

[MCooltA]

Im confused by what you meant above, but i meant to say;

I have found f(x) by ∫f(x,y) dy, with the limits x to ∞.

To then find the E(x) do i ∫x * f(x) dx, with the limits 0 to y?

 

[Ray Vickson]

Im confused by what you meant above, but i meant to say;

I have found f(x) by ∫f(x,y) dy, with the limits x to ∞.

To then find the E(x) do i ∫x * f(x) dx, with the limits 0 to y?

OK, this is equivalent to what I wrote. You really should use different symbols for the different distributions, such as g(x) = int f(x,y) dy and h(y) = int f(x,y) dx, or use f_X(x) instead of g(x) and f_Y(y) instead of h(y).

As to your second question: the region in (x,y) space is {0 <= x <= y}, so yes, for any given x, y goes from x to infinity. However, once y has been "integrated out" it is no longer present, so NO, in EX = int x*f_X(x) dx, x does NOT go from 0 to y---there is no y now!. The variable x goes from 0 to infinity: when x was 5, y went from 5 to infinity, when x was 10 million, y went from 10 million to infinity, etc.

RGV