Can anyone help with the following question please?
The continuous random variables X and Y have the joint probability density function,
f XY (x,y) = 6x, 0 < x < 1, 0 < y < 1-x
Derive the marginal p.d.f for X and Y, and find the marginal expected values for both X and Y.
For the marginal of X, I got 3x(1-x) using the limits of 0 to 1-x
and for Y I got 3, using the limits of 0 to 1.
And using the limits from 0 to 1, I got E(X) = 1/4
My question is what limits, would I used for E(Y), as when I use 0 to 1-x I get E(Y) = (1-x)^3 - 3(1-x)^2
Firstly the marginal pdf of X that you have written is incorrect.
You can check if your odf of X is correct by integrating it from 0 to 1. The pdf should equal to 1.
In this case:
and does not satisfy the above stated property of the pdf.
Therefore, your expected value of X also happens to be incorrect.
To find the expected value of Y, you would integrate:
Thanks, but I thought to find the marginal p.d.f of X, you have to integrate with respect to y.
edit: Sorry my mistake, my integration was incorrect, I got f(x) = -6x(x-1) which also integrate to one.
Lastly your calculation of E[Y] is completely wrong.
you have f(y)= 3. Use this and info in post #2 to find E[Y]
Sorry I meant Var(Y) = (1-x)^3 - 3(1-x)^2
for E(Y) i got 3(1-x)^2 / 2
You need to work on your integration.
Work properly on figuring out the value of E[Y]
Yes, ino, i have nt done it for a while, thanks btw.
Finally, i've got it 3(1-x)^2 / 2
okay so you have"
find now, and then the variance
E(Y^2) = (1-x)^3
Therefore Var(Y) = (1-x)^3 [ 1 - 9(1-x)/4 ]