Finding the joint pdf and then calculating the covariance Let the joint probability distribution of X and Y be $f(x,y)=Te^{-x-y}$ for $0<y<x<\infty$
Find cov(X,Y)
My approach is to first find T. I did this by evaluating T from:
$\int_0^\infty\int_0^xTe^{-x-y}dydx=1$
I got T=2. Is this correct?
Next, I want to use $Cov(X,Y)=E(XY) - E(X)E(Y)$
However, I'm having a problem in getting the marginal distribution of x. I was doing $\int_0^\infty2e^{-x-y}dy$ but this leads to $2e^{-x}$ which is wrong because it is not a valid pdf.
 A: A good way to check one's results is to use a completely different analysis.  I will offer two: statistical reasoning and simulation.
Statistical reasoning
The density is proportional to the product $e^{-x}e^{-y}$, which is instantly recognizable as the product of Exponential densities, and the only additional restriction is $y \lt x$.  That shows us $(y,x)$ have the same distribution as the order statistics of two independent exponential variables.  That is, letting $Z_1$ and $Z_2$ be those variables, set $Z_{(1)} = \min(Z_1,Z_2)$ and $Z_{(2)}=\max(Z_1,Z_2)$.  The vector $(Z_{(1)}, Z_{(2)})$ has the same distribution as $(y,x)$.
Let the expectation of an Exponential variable be $\mu=1$.  Then because $Z_1+Z_2 = Z_{(1)}+Z_{(2)}$ and $Z_{(1)}Z_{(2)}=Z_1Z_2$,
$$E(x+y) = E(Z_{(1)}+Z_{(2)}) = E(Z_1 + Z_2) = E(Z_1)+E(Z_2) = 2\mu=2\tag{1}$$
and
$$E(xy) = E(Z_{(1)}Z_{(2)}) = E(Z_1Z_2) = E(Z_1)E(Z_2) = \mu^2=1\tag{2}.$$
The Exponential distribution is memoryless: this means that $x-y$, conditional on $x \gt y$, also has an Exponential distribution.  Thus
$$E(x)-E(y) = E(x-y) = \mu = 1\tag{3}.$$
The simultaneous linear equations $(1)$ and $(3)$ have the unique solution
$$E(y)=1/2,\ E(x) = 3/2.\tag{4}$$
Plug $(2)$ and $(4)$ into a standard formula for the covariance:
$$\operatorname{Cov}(x,y) = E(xy) - E(x)E(y) = 1 - \frac{3}{2}\frac{1}{2}=\frac{1}{4}.$$
Simulation
It's fast and easy to draw realizations $(x,y)$ from this distribution.  The most obvious way--although it's a little inefficient--is to draw positive $(x,y)$ from the distribution with density $e^{-x-y}$ and discard any for which $y \ge x$.  Since this density factors into $e^{-x}e^{-y}$, it suffices to draw an independent pair from the univariate distribution with density $e^{-x},\ x\gt 0$: the Exponential.  The sample moments for a large set of realizations ought to approximate those of the underlying distribution.  Here is R code to produce and analyze approximately a million such realizations; it takes one second to run.
n <- 2e6
set.seed(17)
a <- matrix(rexp(2*n), ncol=2)
a <- a[a[,2] < a[,1], ]
x <- a[,1]
y <- a[,2]
signif(c(E.xy=mean(x*y), E.x=mean(x), E.y=mean(y), Cov=cov(x,y)), 3)

The output is
E.xy  E.x  E.y  Cov 
1.00 1.50 0.50 0.25 

The values agree exactly with the preceding solution to the three significant figures shown.
A: Thanks to @Greenparker for pointing out where I got it wrong.
So I solved the following to get to the answer:
$f(x)=\int_0^x2e^{-x-y}dy=2e^{-x}-2e^{-2x}$
$E(x)=\int_0^\infty x(2e^{-x}-2e^{-2x})dx=1.5$
$f(y)=\int_y^\infty 2e^{-x-y}dy=2e^{-2y}$
$E(y)=\int_0^\infty 2e^{-2y}=0.5$
$E(xy)=\int_0^\infty\int_0^x2xye^{-x-y}dydx=1$
So, $COV(xy)=1-(1.5)(0.5)=\frac{1}{4}$
