This thesis introduces the concept of Brownian bridge with which is derived the asymptotic distribution of Kolmogorov-Smirnov statistic used for testing whether a random sample comes from a particular population with countinuous distribution. The first chapter is a brief introduction to the concepts of probability theory without which is difficult to understand the subject matter of this paper. Probability space, random variables along with some of its properties and examples of well-known distributions that will be useful later on, are specified in this chapter. Defining the empirical distribution function and its properties brings us the motivation for the main task of this paper. Further, the second chapter defines one of the most important stochastic processes, Brownian motion. Theorems to identify Brownian motion with Gaussian process and to demonstrate that it meets strong Markov property are proved. As a result of strong Markov property, the reflection principle is shown and then applied repeatedly to obtain the proof of Lévy’s triple law. The invariance principle known as Donsker’s theorem is stated. Finally, Brownian bridge and its properties are defined, and it is proved that Brownian bridge is approximately Brownian motion conditioned to be in the neighborhood of the zero at time 1. At the end of this chapter we link Brownian bridge with empirical distribution function. The last chapter of this thesis introduces concepts related to testing of statistical hypotheses, then talks about Kolmogorov-Smirnov statistics defined by the empirical distribution function and displays statistical test in which it is used as a test statistic. Finally we come to the main result which tells us that the Kolmogorov-Smirnov statistic has an asymptotic distribution and that Brownian bridge, and thus Brownian motion give us a way to compute it.