In this report, our goal is to find a way to get some information such as the mean out of high dimensional densities. If we want to calculate the mean we need to calculate integrals, which are difficult to do for high dimensional densities. We cannot use the analytical or classical (deterministic) numerical rules for high dimensional problems for which we want to calculate the mean. These methods take a lot of computational time. To solve this problem we introduced the Markov Chain Monte Carlo (MCMC) method, the method samples from the distribution, and with these samples, we can approximate the mean. Then we explain the theory behind these methods and how we can use it. Then we introduced one MCMC method, in particular, the Metropolis-Hastings Algorithm. We explain how this method works and the theory behind it. From this, we see that the method is very easy to implement and can be used to approximate the mean. Then we approximate the mean for some examples using this method.

In this paper, we want to create a prediction model for target attainment and a scoring system based on these models. The Random Forest model, Logistic Regression model, and Naive Bayes model are employed to achieve this. Classification trees and predictors from the random forest model are used to create the scoring system. The theory behind each model and scoring system is described in detail. The properties of proper scoring rules were checked when making our scoring systems. We have made three models which can reasonably predict target attainment. The logistic regression model performed the best for internal and external validation, with an AUC of 0.84 and 0.80, respectively, whereas the RF model performed slightly worse. The AUC of the random forest differed by 0.01 in both validation sets. The scoring system performed reasonably with an AUC of 0.74. These models and the scoring system perform reasonably well and will help the medical practitioner assess if an ICU patient will attain their beta-lactam antibiotic target.