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

A. A. El-Faheem and Eman A. Mahmoud

**DOI : **10.3844/jmssp.2010.217.220

*Journal of Mathematics and Statistics*

Volume 6, Issue 3

Pages 217-220

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**Abstract**

**Problem statement:** We considered two types of sequences of ordered statistical discrete data i_{1}, i_{2}…i_{n}, 1≤i_{1}≤ i_{2}≤… 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 i_{1}, i_{2}…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.

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**Copyright**

© 2010 A. A. El-Faheem and Eman A. Mahmoud. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.