I am trying to find the join cumulative distribution function of $X$ and $Y$ defined as
$$X = \min(\tau_1, \tau)$$ $$Y = \min(\tau_2, \tau)$$
$\tau$, $\tau_1$ and $\tau_2$ are random variables with exponential distribution of parameters $\lambda$, $\lambda_1$ and $\lambda_2$.
I solved like this
$$F(s,t) = P(X \leq s \land Y \leq t) = P(\min(\tau_1, \tau) \leq s \land \min(\tau_2,\tau) \leq t) = P((\tau_1\leq s \lor \tau \leq s) \land (\tau_2\leq t \lor \tau \leq t)) = \ldots$$
Please, can you give me a hint on how to go on? I got stucked.
The solution is $$F(s,t) = P(X\leq s \land Y \leq t) = 1 - P(X > s \lor Y > t) =$$ $$= 1 - \left[P(X>s) + P(Y>t) - P(X > s \land Y > t)\right] =$$ $$ =1 - \left( e^{-(\lambda_1 + \lambda )s} + e^{-(\lambda_2 + \lambda )t} - e^{-\lambda_1 s - \lambda_2 t - \lambda \max(s,t)}\right) = 1 - e^{-(\lambda_1 + \lambda )s} - e^{-(\lambda_2 + \lambda )t} + e^{-\lambda_1 s - \lambda_2 t - \lambda \max(s,t)}$$