The Probability and the Asymptotic Distribution Functions of Increasing Ordered Statistical Data

Problem statement: We considered two types of sequences of ordered statistical discrete data i₁, i₂…i_n, 1≤i₁≤ i₂≤… i_n≤n and i_j = 1,2,…,n, j = 1,2,…,n. The first type is the order statistics of a sample of size n taken from the uniform discrete distribution on the set S = {1,2,…,n}, n  ∈ N⁺. The other type is the increasing ordered word of length n taken from the set S = {1,2,…,n}, n ∈ N⁺. We studied a common property of both types of samples namely the number of elements i_j = j, i_j = 1,2,…,n, j = 1,2,…,n. i.e., we find the number of integers in each sequence i₁, i₂…i_n satisfying the condition C: i_j = j, for both types. We obtain the probability distribution functions as well as the asymptotic distributions of the random variables representing the number of integers satisfying condition C. Approach: We employed combinatorial tools to obtain the number of samples having j elements satisfying condition C in both types, j = 1,2,…,n. Results: For large n ∈ N⁺, we found that the expected value of the number of samples of the second type, satisfying condition C, is much larger than that of the first type. Conclusion: The result can be used to distinguish between these two data.