How is $(X,X+Y)$ distributed when $X$ and $Y$ are independent random variables?

Question

Let $X,Y$ be two independent random variables on the same probability space. The distribution of $X$ is $F^X$ and the distribution of $Y$ is $F^Y$.

The distribution of $X+Y$ is $(F^X * F^Y)$ because of independence. How is the vector $(X,X+Y)$ distributed ?

Answer 1

Comment continued. Maybe an example will give you an idea how to proceed. Consider $X\sim\mathsf{Exp}(1)$ and, independently, $Y\sim\mathsf{Exp}(1).$ Also, let $S = X+Y.$ Tnen $E(X) = E(Y) = 1, E(S) = 2;$ $Var(X) = Var(Y) = 1;$ $Var(S) = Var(X)+Var(Y)=1+1=2;$ $Cov(X,S) = Cov(X,X+Y) = Var(X) = 0 = 1.$ $\rho_{X,S} = Cor(X,S) = \frac{Cov(X,S)}{SD(X)SD(S)} = \sqrt{2}/2.$

Finally, $S \sim\mathsf{Gamma}(\mathrm{shape}=2,\mathrm{rate}=1),$ perhaps shown by multiplying MGF's.

Simulation, by sampling a million points from $(X,S)$ in R, sample statistics should match corresponding population parameters to a couple of decimal places.

set.seed(2021)
x = rexp(10^6);  y = rexp(10^6)
s = x + y
summary(x); var(x)
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
 0.000001  0.287486  0.692905  0.999743  1.384461 13.680783 
[1] 1.001571  # aprx Var(X) = 1
summary(s);  var(s)
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
 0.001398  0.961369  1.678681  1.999379  2.692112 18.659487 
[1] 1.996201  # aprx Var(S) = 2
cov(x,s)
[1] 1.000792  # aprx Cov(X,S) = 1
cor(x,s)
[1] 0.7077843 # aprx Cor(X,S) = 0.7072
1/sqrt(2)
[1] 0.7071068

Marginal distributions are approximated by histograms and marginal density functions are shown:

par(mfrow=c(1,2))
 hist(x, prob=T, br=30, col="skyblue2", main="Dist'n of EXP(1)")
  curve(dexp(x), add=T, lwd=2, col="orange")
 hist(s, prob=T, br=30, col="skyblue2", main="Dist'n of GAMMA(2,1)")
  curve(dgamma(x,2,1), add=T, lwd=2, col="orange")
par(mfrow=c(1,1))

For the general case, I am not sure how you are expected to show the joint distribution. For my example, a scatterplot of the first 50,000 points sampled suggests the joint distribution. Notice that the support of the joint distribution in my example lies entirely in the first quadrant above the 45-degree line. [Perhaps, you're expected to show the jour density as $f_{(X,S)}(x,s) = f_X(s)f_{Y|X=x}(y|x).]$

X = x[1:50000];  S = s[1:50000]
plot(X,S, pch=".", main="Scatterplot of Joint Dist'n")