Independent Exponential Random Variables

Question

I am currently trying to figure out a problem and it is using notation that I have never seen before so I am pretty stuck, any suggestions would be greatly appreciated!

Let $X, Y, Z$ be independent exponential random variables with the same mean, $σ$. Find the value of $σ$ so that $\Pr[\max(X,Y,Z)>1]=0.05$.

Any help with how to solve this problem or leading down the right path would be awesome as I am not to sure where to even start!

EDIT: So I have followed what André Nicolas has said and got a $σ$ of $0.25$. However I am still not sure where he got this formula from: The probability they are all $\le a$ is $(1-e^{-a/\sigma})^3$.

I cant find anything like this is my notes or textbook, could any reference where he got this?

Answer 1

A start: The probability that $\max(X,Y,Z)\gt 1$ is $1$ minus the probability they are all $\le 1$.

The probability they are all $\le a$ is $(1-e^{-a/\sigma})^3$.

Answer 2

$$p(0\le X\le x)=1-e^{-\lambda X}$$
where $\lambda$ is $1/\text{mean}$. When $x=1$, the mean is $a$, and so the equation becomes $1-e^{-1/a}$.