Given U,V independent exponential R.Vs. Find the law of min(U,V).
I understand some of the basics, but I need exact formal proof at the beginning since I've seen very few of these questions. I will write the beginning which I hesitate of. Could you correct my mistakes? Or approve my 100% correctness.
My solution:
U and V are Random Variables. Thus we are looking for
$P\{\{t<U\} \bigcap \{t<V\}\}$
U, V are independent. Hence
$=P\{t<U\}\bigcap P\{t<V\}$
Which is in turn:
$=F_U(t)F_V(t)$
And then we calculate and differentiate.
Did I write it wrong somewhere? Is everything correct?