# How to Prove that the minimum of two exponential random variables is another

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

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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 $\lambda+\mu$.

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