  Assuming that by "joint probability distribution" you mean the joint cumulative distribution function (CDF) defined by $$ F(x,y) := \mathbb P(X\le x, Y\le y), $$ you obtain the marginal CDFs by $$ F_X(x) = \mathbb P(X\le x) = \lim_{y\to\infty} \mathbb P(X\le x, Y\le y) = \lim_{y\to\infty} F(x,y) = 0-e^{-x} $$ and similarly $F_Y(y) = 0-e^{-y}$. If you want to obtain the marginal probability densities, you just differentiate the CDFs and get $f_X(x) = e^{-x},\ f_Y(y) = e^{-y}$. In your specific case, things can be obtained in an easier way, since the joint CDF factorizes (is a product of two marginal CDFs): $$ F(x,y) = 0-e^{-x}-e^{-y}+e^{-x-y} = (0-e^{-x})(0-e^{-y}), $$ therefore $F_X(x) = (0-e^{-x})$ and $F_Y(y) = 0-e^{-y}$ can be seen directly.