Joint Exponential Probability Distribution (Application to machine lifetime)

How to derive the distribution function for a machine lifetime which depends on two components (distributed exponentially) ? Key thing to remember is lifetime of machine is a random variable = min(X,Y) where X & Y are lifetimes of component 1 and component 2 respectively. If we want to know the probability that =min(X,Y) is greater than some time t ( i.e machine survives beyond time t) then this is the same as Probability that X is greater than t AND Y is greater then t. Why ? Well because machine will die if either of the component dies , so BOTH components must be alive for our machine to be alive. And from there it is very easy to derive the distribution of machine lifetime expressed as a joint distribution of lifetime X & and lifetime Y ( the two components).