3.) Let X and Y be continuous random variables with joint density function
How would you calculate the covariance/correlation coefficient with those bounds? thank you for your help.
This is basic calc 3.
YOU must draw the region.
First draw the lines y=x, y=2x and x=1.
Then find the three vertices, (0,0), (1,1) and (1,2).
This rectangle is easier to integrate via dydx than dxdy.
Once you have the bounds the three integrals are all the same except for the integrand.
IT's better to look at this two dimensionally and NOT obtain the marginals.
Obtaining the marginals forces you to do a particular order of integration
and I prefer dydx in all three cases in this example.