Marginal Distributions of given CDF

Question

I have the CDF given by :

$$F(x_1, x_2) = e^{-(-x_1-x_2)^{1/\beta}}$$

with $x_1,x_2 \leq 0$ and $\beta \geq 1 .$

I need to find the marginal distribution functions. However when I try to apply the limit to infinity for any of these two random variables I get something not determined, i.e.

$$F_{X_1}(x_1) = \lim_{x_2 \to \infty} e^{-(-x_1-x_2)^{1/\beta}} "=" e^{-(-\infty)^{1/\beta}} "=" e^{-\infty(-1)^{1/\beta}}$$

which, dependent on $\beta$, is either a complex infinity or a real infinity.

Does somebody see what I am missing here?

Thanks!

Answer 1

$F_{X_1}(x_1)=F_{X_1X_2}(x_1;0)=e^{-(-x_1)^{\frac{1}{\beta}}}\mathbb{1}_{(-\infty;0)}(x_1)+\mathbb{1}_{[0;\infty)}(x_1)$

and similarly for $X_2$