I am having a very hard time understanding probability problems involving max and min distributions, for example:
For $n$ i.i.d. exponential random variables, find $\Bbb P(X_1=\text{min}(X_1,...,X_n))$.
Can someone explain to me what exactly $\text{min}(X_1,...,X_n)$ refers to? All I can think is the probability that $X_1$ equals the smallest $X_i$, but if they are all identical they are all the same size. Please help!
Hope this example helps. Let's say you use two identical alarm clocks to set an alarm to wake you up. Each alarm goes off at a random time which follows the same probability distribution. Let's call these times for the first and the second clock $X_1$ and $X_2$. Now $X_1$ and $X_2$ are i.i.d., and $\mathbb P(X_1 = \min (X_1, X_2))$ is equivalent to asking "What is the probability that the first clock wakes you up?"
The answer is, of course, $0.5$. This is because the two clocks are exactly the same, so each has an equal chance of going off first. Similarly, in the case of $n$ variables, the answer is $\frac 1 n$.