# random variables – Independence of two events, \${S

I was working through J.R. Norris Markov Chains and got stuck at exercise 2.3.1.
The question is the following:

Suppose $$S, T$$ are independent exponential random variables of parameters $$alpha$$ and $$beta$$ respectively. What is the distribution of $$min{S,T}$$? What is the probability that $$S leq T$$? Show that the two events $${S < T}$$ and $${min{S,T} geq t}$$ are independent

I was able to prove:

1. $$min{S,T} sim exp(alpha + beta)$$
2. $$mathbb{P}(S leq T) = frac{alpha}{alpha + beta}$$

Now for the last problem, I have tried to prove the following ($$Z := min{S,T})$$
$$mathbb{P}(Z geq t, S leq T) = mathbb{P}(Z geq t)mathbb{P}(S leq T).$$

Now the left hand side gives

$$mathbb{P}(Z geq t, S leq T) = mathbb{P}(min{S,T} geq t mid S leq T)mathbb{P}(S leq T) = mathbb{P}(S geq t)mathbb{P}(S leq T).$$

But since $$mathbb{P}(S geq t) neq mathbb{P}(Z geq t)$$, I would conclude that the two events in question are not independent.

My questions are:

1. What am I doing wrong?
2. How do you actually solve this question?

Thanks in advance!

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