Hierarchical spatio-temporal models are used for modeling space-time responses and temporally and spatially correlations of the data is considered via Gaussian latent random field with Matérn covariance function. The most important interest in these models is estimation of the model parameters and the latent variables, and is predict of the response variables at new locations and times. In this paper, to analyze these models, the Bayesian approach is presented. Because of the complexity of the posterior distributions and the full conditional distributions of these models and the use of Monte Carlo samples in a Bayesian analysis, the computation time is too long. For solving this problem, Gaussian latent random field with Matern covariance function are represented as a Gaussian Markov Random Field (GMRF) through the Stochastic Partial Differential Equations (SPDE) approach. Approximatin Baysian method and Integrated Nested Laplace Approximation (INLA) are used to obtain an approximation of the posterior distributions and to inference about the model. Finally, the presented methods are applied to a case study on rainfall data observed in the weather stations of Semnan in 2013.

Among all statistical distributions, standard normal distribution has been the most important and practical distribution in which calculation of area under probability density function and cumulative distribution function are required. Unfortunately, the cumulative distribution function of this is, in general, expressed as a definite integral with no closed form or analytical solution. Consequently, it has to be approximated. In this paper, attempts have been made for Winitzki's approximation to be proved by a new approach. Then, the approximation is improved with some modifications and shown that the maximum error resulted from this is less than 0.0000584. Finally, an inverse function for computation of normal distribution quantiles has been derived.

The collective risk model of insurance company with constant initial capital when process of claims number have the poisson distribution with constant rate is considered. For computing the infinite time ruin probability the stochastic processes and differential equations are used. Also a formula is obtained to compute the Lundberg approximation in finding the approximate of infinite time ruin probability based on the distribution function of claims number. The numerical examples to illustrate these results are given and showed that for any value of initial capital the approximate of our infinite time ruin probability is closer to its real value rather than the ruin probability computed by other authors and has less error.

An important issue in many cities is related to crime events, and spatio–temporal Bayesian approach leads to identifying crime patterns and hotspots. In Bayesian analysis of spatio–temporal crime data, there is no closed form for posterior distribution because of its non-Gaussian distribution and existence of latent variables. In this case, we face different challenges such as high dimensional parameters, extensive simulation and time-consuming computation in applying MCMC methods. In this paper, we use INLA to analyze crime data in Colombia. The advantages of this method can be the estimation of criminal events at a specific time and location and exploring unusual patterns in places.