Let $X \sim$ Exponential (rate = $\theta$) and $Y \sim$ Exponential (1) be independent random variables.
Define $Z \equiv \min\{X,Y\}$
I have two questions:
(1) What is the joint distribution of $X$ and $Z$?
(2) What is the joint distribution of $Y$ and $Z$?
Obviously these are both interrelated questions, as the answer for one probably yields the precise method to answer the other. I have started by trying to compute the joint cdf of $X$ and $Z$ and then taking two partial derivatives:
$$F_{X,Z}(x,z) = P(X \leq x, Z \leq z) = P(X \leq x \wedge z, X \leq L) + P(X \leq x, L \leq z, L < X) \\ = \int_{0}^{x \wedge z}\int_{x_1}^\infty \theta e^{-\theta x_1} e^{-l_1}dl_1 dx_1 + \int_{0}^{x}\int_{0}^{x_1 \wedge z}\theta e^{-\theta x_1} e^{-l_1}dl_1 dx_1$$
but I am having trouble computing the integrals. Is this the right approach? If so, maybe there is an easier way? If not, what is the right way?
If $x \le z$ then \begin{align} P(X \le x, Z \le z) = P(X \le x) = 1 - e^{-\theta x}. \end{align} If $x \ge z$ then \begin{align} P(X \le x, Z \le z) &= P(X \le z, Z \le z) + P(z < X \le x, Y \le z) \\ &= P(X \le z) + P(z < X \le x) P(Y \le z) \\ &= (1 - e^{-\theta z}) + (e^{-\theta z} - e^{-\theta x}) (1 - e^{-z}) \end{align}
This joint distribution does not have a density with respect to Lebesgue measure because $P(X=Z) > 0$.