Random projections is a technique used primarily in dimension reduction, in order to estimate distances in data. They can be thought of a linear transformation mapping a data matrix X to a lower dimensional space, where distances are preserved in expectation. However, the preservation of distances can be thought of a stepping stone to some eventual goal, such as classification, hypothesis testing, information retrieval, or even reconstructing principal components of data. In this thesis, I will give a background of the basic random projection algorithm. Next, I then look at the structure of random projection matrices and propose modifications to result in a more accurate estimation of distances, which would help in information retrieval and reconstruction of principal components. Finally, I show that it is possible to juxtapose the use of Monte Carlo variance reduction methods with random projections to improve the accuracy of distance estimates, which can then be used in an algorithm or procedure of the users' choice. Theoretical justifications are given, and empirical results are shown with synthetic data, and experiments from publicly available datasets.