Probability that a random variable is among the top k out of n when ordered Suppose $X_1,X_2,\ldots,X_n $ are $n$ i.i.d. random variables with a continuous distribution $F(x)$ and density function $f(x)$.
What is the probability distribution that any given $X_i$ is among the top $k$ largest of the $n$ $X$'s?
For example, there are $10$ individuals. Each of them draw a random number from a distribution $F(x)$ with density function $f(x)$. The draws are i.i.d. What is the probability that individual $i$'s draw will be among the top $3$ largest numbers (i.e. either top, second or third)?
Many thanks.
 A: $X_1,\ldots,X_n$ are the $n$ i.i.d. random variables.
Let $X_{(1)}<\cdots< X_{(n)}$ be the order statistics, i.e. the same random variables sorted.  (By continuity of the c.d.f., we need not write "$\le$".)
Then $X_i=X_{(j)}$.  Given $i$, what is the distribution of $j\text{ ?}$
Let $Y_1,\ldots,Y_n$ be $X_{\sigma(1)},\ldots,X_{\sigma(n)}$, where $\sigma$ is some permutation.
Lemma: The joint distribution of $Y_1,\ldots,Y_n$ is the same as the joint distribution of $X_1,\ldots,X_n$.
Hence $\Pr(Y_i = Y_{(j)})=\Pr(X_i=X_{(j)})$.  But $\Pr(Y_i=Y_{(j)})$ is the probability that $X_{\sigma(i)}$ is in the $j$th position when sorted.  Thus the probability that $X_{\sigma(i)}$ is in the $j$th position is the same as the probability that $X_i$ is in the $j$th position.
This works regardless of which permutation $\sigma$ is, so it shows that every index has the same probability of being in the $j$th position.  This is true for every value of $j$.
Hence the rank of $X_i$ in the sorting is uninformly distributed on the set $\{1,\ldots,n\}$.  It probability of being in any particular subset is the size of that subset divided by $n$.
