Exercise 9.1: The Gaussian distribution

1. Use the Metropolis algorithm to generate a Gaussian distribution $P(x)=A \exp{(-x^2/2\sigma ^2)}$. Is the numerical value of the normalization constant $A$ relevant? Determine the qualitative dependence of the acceptance ratio and the equilibrium time on the maximum step size $\delta$. One possible criterion for equilibrium is that $\langle x^2 \rangle \approx \sigma ^2$. For $\sigma = 1$, what is a reasonable choice of $\delta$? (choose $x_0 = 0$.)

2. Plot the asymptotic probability distribution generated by the Metropolis algorithm.