Hamdy N.Abd-ellah حمدي نور الدين عبد الله Department of Mathematics, Faculty of Science, Assiut University Assiut, Egypt جامعة أم القرى قسم الرياضيات الكلية الجامعية بمكة المكرمة

Information Geometry of Random Walk Distribution 4

Abstract: Information geometry (Geometry and Nature) as emerged from the study of invariant properties of the manifold of probability distributions. It is regarded as mathematical sciences having vast developing areas of applications as well as giving new trends in geometrical and topological methods. Information geometry has many applications which are treated in many different branches, for instance, statistical inference, linear systems and time series, neural networks and nonlinear systems, linear programming, convex analysis and completely integrable dynamical systems, quantum information geometry and geo-metric modeling. 5

Here, we give a brief account of information geometry and the deep relationship between the deferential geometry and the statistics[1; 4; 5; 10; 11] :The parameter space of the random walk distribution using its Fisher's matrix is defined. The Riemannian and scalar curvatures of the parameter space are calculated. The differential equations of the geodesic are obtained and solved. The relations between the J- divergence and the geodesic distance in that space are found. 6

1. Geometrical and statistical preliminaries: The concept of a metric and a tensor, in generality, is well known, we turn now to statistical application. Therefore, we give some formulas and definitions of the geometrical and statistical concepts which are necessary for later use. 7

Definition1.1[10]: A statistical model is a family S of probability distributions P of random variables which is smoothly parameterized by a finite number of real parameters i as the following 8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

27

30

31

32

33

34

35

36

37

38

39

40

43

44

45