Generation of (pseudo-)Random Numbers - Theoretical Curve To The Histogram - Statistics Assessment Answer

Internal Code: 1AHCIA

Statistics Assessment Answer

TASK: 1 Generation of (pseudo-)random numbers Select the distribution of a random variable X with finite expectation and variance (you can use the book by Devroye as a source) and write an algorithm in R that generates random numbers distributed according to the selected distribution. Plot a histogram of the random numbers and compare it with the theoretical distribution by superimposing a theoretical curve to the histogram. You cannot use the same distributions and programs that are online in Canvas. 2 Markov Chains and Markov Chain Monte Carlo Consider a homogeneous Markov chain Zn whose state space has three states and with the following transition probability matrix, Choose the values of the parameters ?,?,?,?, ",?,?,?,? so that they are in the interval [0, 1) and the Markov chain is irreducible and aperiodic and clearly explain why your choice leads to an irreducible and aperiodic Markov chain. Compute the invariant distribution of your Markov chain. Write a Monte Carlo program in R that simulates your Markov chain and compare the empirical distribution you obtain by Monte Carlo simulations with the theoretical invariant distribution. Using the theorems presented during the course, explain why the Monte Carlo algorithm works. 3 Select at least one of the following projects: 3.1 Ising model Consider an Ising model with five spins on a graph of your choice in which each spin is connected with at least another spin (please do not use the cyclic graph presented in the lecture notes). Write the Hamiltonian for your model. Compute the partition function, the free energy, the entropy, the energy and the absolute value of the magnetisation of your model. Write a Metropolis Monte Carlo simulation of your model and check that it gives the correct values for the energy per spin and the absolute value of the magnetisation per spin. To this purpose compare the plot the results of the Monte Carlo simulations against your theoretical results as a function of temperature (or inverse temperature). 3.2 Random walk Use the random variable you introduced in point 1 to define a random walk as follows. Let {Xi}Ni=1 be a sequence of independent and identically distributed random variables with cumulative distribution function FX (u) = P(X ? u). Consider the random variable Theoretically compute the cumulative distribution function FZN (u) = P(ZN ? u) of ZN . Write a Monte Carlo program in R that generates samples of ZN for N ? 1. Choose a value of N strictly larger than 2 and plot a histogram of ZN comparing it with your theoretical calculation. According to the central limit theorem if E(X) < ? and Var(X) < ?, the variable.

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