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‎Given the importance of Markov chains in information theory‎, ‎the definition of conditional probability for these random processes can also be defined in terms of mutual information‎. ‎In this paper‎, ‎the relationship between the concept of sufficiency and Markov chains from the perspective of information theory and the relationship between probabilistic sufficiency and algorithmic sufficiency is determined‎. 

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In this paper we explain a necessary and sufficent condition for independence between any arbitrary statistics with sufficient statistics which is also maximum likelihood estimator in a general‎
‎exponential family with location and scale parameter namely generalized normal distribution‎. ‎At the end‎, ‎it is shown that the converse is true except in the asymptotic cases‎.

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‎In this paper‎, ‎after introducing exponential family and a history of work done by researchers in the field of statistics‎, ‎some applications of this family in statistical inference especially in estimation problem‎,‎statistical hypothesis testing and statistical information theory concepts will be discussed‎.

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‎In this paper after introduce Ito integral we discuss filtering problem‎. ‎In filtering problem there are two stochastic differential equations (system and observation) that given the observations we must find the best estimate for the random process of the system based on these observations‎. ‎At last we give some useful examples‎.

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‎Information inequalities have many applications in estimation theory and statistical decision making‎. ‎This paper describes the application of an information inequality to make the minimax decision in the framework of Bayesian theory‎. ‎In this way‎, ‎first a fundamental inequality for Bayesian risk is introduced under the square error loss function and then its applications are expressed in determining asymptotically and locally minimax estimators in the case of univariate and multivariate‎. ‎In the case that the parameter components are orthogonal‎, ‎the asymptotic-local minimax estimators are obtained for a function of the mean vector and the covariance matrix in the multivariate normal distribution‎. ‎In the end‎, ‎the bounds of information inequality are calculated under a general loss function‎.

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Copula functions are  useful tools in modeling the dependence between random variables, but most existing copula functions are symmetric, while in many applications, asymmetric joint functions are required. One of these applications is reliability modeling, where asymmetric joint functions can explain different tail dependencies and provide a better model. Therefore, the theory of constructing asymmetric copula functions that can model a wider range of data has been developed. In this research, while reviewing the methods of creating asymmetric copula functions that can provide various tail dependencies, these functions are used to estimate the two-dimensional reliability of data on the age an usage of Rana and Dana cars.