MATH 351 Homework 08

1. Use inversion sampling to sample $N$ random numbers from the logistic distribution. The logistic distribution has cumulative distribution function $$F(x) = \frac{1}{1 + e^{-(x-\mu) / s}}$$ You pick values for the parameters $\mu \in \mathbb{R}$ and $s > 0$. Confirm your answer in at least two different ways.
2. Use inversion sampling to sample $N$ random numbers from the Pareto distribution. The Pareto distribution has cumulative distribution function $$F(x) = 1 - \left(\frac{x_m}{x}\right)^{\alpha}$$ for $x > x_m$. You pick values for the parameter $x_m > 0$ and $\alpha > 0$. Confirm your answer in at least two different ways.
3. Use the Metropolis Algorithm to sample $N$ random numbers for the parameter $\sigma$ from the Rayleigh distribution based on a random sample of Rayleigh data $\mathbf{x}$. The idea here is to learn about the value of $\sigma$ by approximating expectations of a distribution for $\sigma$ based on a sample of data $\mathbf{x}$. The Rayleigh distribution has probability density function $$f(x | \sigma) = \frac{x}{\sigma^2}e^{-x^2 / (2\sigma^2)}$$ for $x > 0$ and $\sigma > 0$. Here's some code to generate data from a Rayleigh distribution for a specified value of $\sigma$, which you should choose.

   import scipy.stats as st
   x = st.rayleigh(scale = 5).rvs(size = 501)