Suppose that $X_1,\cdots,X_n$ are i.i.d. continuous random variables. $X_k$ is called good if we have $X_i < X_k$ for all $i<k$.
1) Find the probability that $X_k$ is good.
2) Find the expected value of the number of good random variables. Show that this value approaches infinity when $n \to \infty$.
Please do not give me a full solution first. Just hints. My major issue with this problem is that I do not understand what the question is asking mathematically. So, a random variable is just a function from some probability space to real numbers. What kind of probability density are we assuming on the space of random variables? What is our probability space here?
If we sort the variables $X_1\dots X_k$ in increasing order, there are $k!$ possible orders. Only the ones where $X_k$ comes last make $X_k$ good. There are $(k-1)!$ such arrangements. Since, by symmetry, all orders are equally likely, the probability that $X_k$ is good is $${(k-1)!\over k!}={1\over k}$$
We ignore the possibility that two of the variables have the same value. Since the X_i are continuous r.v.'s, the probability that two of them are equal is $0.$
The expected number of good $X_k$ is just $$\sum_{k=1}^\infty P(X_k\text{ is good})=\sum_{k=1}^\infty{1\over k} = \infty$$