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There are many researches on the vibration behavior of the multi-phase flow in the pipes. However, there isn’t any general statistical study on the dynamic response of such systems. Therefore in this paper, at the first step, the nonlinear equation governing the transverse vibration of the pipe is derived using the Hamilton's principle. The nonlinearity in the system is induced by considering large deflections. The interaction between the pipe and the multi-phase fluid flow and the resultant uncertainty is modeled by random excitation which is produced by using normal distribution function. After extraction of the governing equation and discretizing it by the Galerkin method, the equations are solved numerically. The statistical parameters of the response have been extracted by Monte-Carlo simulation. With studying on the deflection of one point on the pipe and also considering corresponding upper and lower limit band (confidence interval), extended results of uncertainties effects have been obtained. The results show that with increasing the velocity of the fluid, the uncertainty of the response is decreasing. Also by considering nonlinear model, the probabilities of failure are increased.

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Three k-tree distance and fixed-sized plot designs were used for estimating tree density in sparse Oak forests. These forests cover the main part of the Zagros mountain area in western Iran. They are non-timber-oriented forest but important for protection purposes. The main objective was to investigate the statistical performance of k-tree distance and fixed-sized plot designs in the estimation of tree density. In addition, the cost (time required) of data collection using both k-tree distance and fixed-sized plot designs was estimated. Monte-Carlo sampling simulation was used in order to compare the different strategies. The bias of the k-tree distance designs estimators decreased with increasing the value of k. The Moore’s estimator produced the smallest bias, followed by Kleinn and Vilcko andthen Prodan. In terms of cost-efficiency, Moore’s estimator was the best and Prodan’s estimator was superior to Kleinn and Vilcko’s estimator. Cost-efficiency of k-tree distance design is related to three factors: sample size, the value of k, and spatial distribution of trees in a forest stand. Moore’s estimator had the best statistical performance in terms of bias, in all four-study sites. Thus, it can be concluded that Moore’s estimator can have a better performance in forests with different tree distribution.