To simulate random events using computer programs, we often use probability distributions. In this task, we will sample random variable X from Log-Normal distribution. The generated numbers will be then plotted on a histogram, which will be compared to the theoretical distribution by overlaying a theoretical curve on the histogram. Subsequently, to quantitatively assess the goodness of fit between the empirical and theoretical distributions, we will use a goodness-of-fit method as a function of sample size.
In this task we are asked to simulate a three state Markov chain using Monte-Carlo simulations. We will use MCMC techniques to derive the invariant distribution of the transition probabilities associated with the three states.
In this section we will define a random walk using the Log-Normal distribution selected in the first section. The random walk is defined as the sum N random variables drawn from the Log-normal distribution, i.e., ZN = N∑ i=1 Xi. After that, we will calculate the cumulative distribution function for ZN theoretically and compare it with the empirically derived CDF which will be generated using Monte Carlo simulations.