Show that the CDF of $Z$ is exponential with parameter $λX + λY$

Question

Two runners run a long distance course in times $X$ and $Y$ , which are independent and exponentially distributed with parameters $\lambda_X$ and $\lambda_Y$ , respectively. The winner’s time is $Z = min\{X, Y\}$. Show that the CDF of $Z$ is exponential with parameter $\lambda_X + \lambda_Y$

Answer 1

There's a trick for this one.
$P(Z>t)=P(X>t,Y>t)=P(X>t)P(Y>t)=e^{-\lambda x}e^{-\lambda y}=e^{-\lambda(x+y)}$