How to prove that minimum of two exponential random variables is another exponential random variable?

Question

How can I prove that the minimum of two exponential random variables is another exponential random variable, i.e. Z = min(X,Y)

Answer 1

Note that you must assume that $X$ and $Y$ are independent, otherwise the result is easily seen to be false.

There is a constant $\lambda$ such that $P(X \geq t)=e^{-\lambda t}$ for every $t>0$.

There is a constant $\mu$ such that $P(Y \geq t)=e^{-\mu t}$ for every $t>0$.

Then for every $t>0$ we have

$$ P(Z \geq t)=P(X\geq t,Y\geq t)=P(X\geq t)P(Y\geq t)=e^{-(\lambda+\mu)t} $$

So $Z$ is an exponential random variable with parameter $\lambda+\mu$.