# What is the probability that $X_i$ is the $k^{th}$ order statistic in consecutive trials?

Consider $$n$$ r.vs $${X_1, X_2,...,X_n}$$. Each is i.i.d drawn from some distribution $$f(.)$$. What is the probability that $$X_i$$ is the $$k^{th}$$ order statistic in any two consecutive trials?

• What do you mean with "in two consecutive trials"? When you say "$X_i$ is the kth order statistic", you mean "there are exactly $k-1$ elements of $\{X_1,\dots,X_n\}$ that are smaller than $X_i$"? Can you assume that the distribution is continuous, so that ties happen with probability 0? – Federico Poloni Jun 16 at 16:22
• Assume $X_{i,j}$ are continuous random variables, so no ties, and are i.i.d. across i=1 ..n and j=1,2.. Let $X_{i,1}$ be the random variables in the 1st trial, and $X_{i,2}$ be the random variables in the 2nd trial,. Assume exactly 2 trials are conducted, each consisting of n i.i.d. variables being drawn..Specify a value of $i$. Let k be the value such that $X_{i,1}$ in the 1st trial is the kth order statistic of the first trial. Then the probability that $X_{i,2}$ is the kth order statistic of the 2nd trial is $1/n$. – Mark L. Stone Jun 17 at 23:20
• Dear Federico, Let us say there are $n$ r.v.s each of which can take some values between range [a,b]. Yes, $k^{th}$ statistics mean there are exactly $k-1$ smaller elements than $X_i$. So what is the probability that in two consecutive trials $X_i$ will have the same position in the order/arrangement? – Suvadip Batabyal yesterday