Available statistical methodologies focus more on accommodating continuous variables, however recently dealing with count data has received high interest in the statistical literature. In this dissertation, we propose some statistical approaches to investigate linear and nonlinear dependencies between two discrete random variables, or between a discrete and continuous random variables. Copula functions are powerful tools for modeling dependencies between random variables. We derive copula-based population version of Spearman’s rho when at least one of the marginal distribution is discrete. In each case, the functional relationship between Kendall’s tau and Spearman’s rho is obtained. The asymptotic distributions of the proposed estimators of these association measures are derived and their corresponding confidence intervals are constructed, and tests of independence are derived. Then, we propose a Bayesian copula factor autoregressive model for time series mixed data. This model assumes conditional independence and shares latent factors in both mixed-type response and multivariate predictor variables of the time series through a quadratic timeseries regression model. This model is able to reduce the dimensionality by accommodating latent factors in both response and predictor variables of the high-dimensional time series data. A semiparametric time series extended rank likelihood technique is applied to the marginal distributions to handle mixed-type predictors of the high-dimensional time series, which decreases the number of estimated parameters and provides an efficient computational algorithm. In order to update and compute the posterior distributions of the latent factors and other parameters of the models, we propose a naive Bayesian algorithm with Metropolis-Hasting and Forward Filtering Backward Sampling methods. We evaluate the performance of the proposed models and methods through simulation studies. Finally, each proposed model is applied to a real dataset.