= Solution
Compute all three <probabilities>. From the preceding part, $\mathbb P(A)=e^{-6}$. Integrating the <joint probability density function> over $u>v$ gives
$$
\mathbb P(B)=\int_0^\infty\int_v^\infty2e^{-u-2v}\,du\,dv
=\int_0^\infty2e^{-3v}\,dv=\frac23.
$$
On $A\cap B$ one has $v>2$ and $u>v$, so
$$
\mathbb P(A\cap B)=\int_2^\infty\int_v^\infty2e^{-u-2v}\,du\,dv
=\frac23e^{-6}=\mathbb P(A)\mathbb P(B).
$$
Therefore \b[$A$ and $B$ are independent events]. The minimum waiting time and the identity of the first clock are independent for <competing exponential clocks>; unequal rates change the winner <probabilities> but do not destroy that <independence>.
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