Due 23:59 Sunday, 2022-06-19
Due 23:59 2022-06-19
Write a linear congruential generator \[ \begin{split} x_{i+1} &= (a x_i + c) \mod m, \quad i=1, 2, \dotsc \\ u_i &= x_i / m. \end{split} \] for \(m = 2^{32}\), \(a = 1103515245\), \(c = 12345\), and for \(m = 2048\), \(a = 1229\), \(c = 1\) respectively. Compare the two random number generators.
In case the inverse CDF \(F^{-1}\) is not exclicitly available, can you implement the inverse CDF method for random number generation? Assume that the pdf of the distribution exists. Implement your method.
As an alternative to the Box-Muller or Marsaglia method, consider the following method of sampling a normal random variable using the absolute value \(X=|Z|\) of a standard normal random variable \(Z\). 1. Find the probability density function \(f_X\) of \(X\). 2. If \(g\) is the probability density function of an exponential random variable with mean \(1\), find the smallest \(c > 0\) such that \[ \frac{f_X(x)}{g(x)} \le c, \quad \text{for all } x > 0. \] 3. Propose an algorithm that generate \(X\), the absolute value of the standard normal random variable \(Z\). 4. How would you generate \(Z\)? 5. Implement your algorithm and test. 6. How efficient is your algorithm?
Consider testing the hypotheses \(H_0: \lambda=2\) versus \(H_a: \lambda>2\) using 25 observations from a Poisson(\(\lambda\)) model. Rote application of the central limit theorem would suggest rejecting \(H_0\) at \(\alpha=0.05\) when \(Z > 1.645\), where \(Z = \frac{\bar{X}-2}{\sqrt{2/25}}\).
Estimate the size of this test (i.e., the type I error rate) using three Monte Carlo approaches: standard, antithetic, (unstandardized) importance sampling. Provide a confidence interval for each estimate. Discuss the relative merits of each variance reduction technique.
For the importance sampling approach, use a Poisson envelope (\(h\) in the course notes) with mean equal to the \(H_0\) rejection threshold, namely \(\lambda = 2.4653\).
Draw the power curve for this test for \(\lambda \in [2.2,4]\), using the same three techniques. Provide pointwise confidence bands in each case. Discuss the relative merits of each technique in this setting.