Exercise 4

Rejection Algorithms for the Standard Normal Random Variable

This exercise focuses on using rejection sampling to generate standard normal random variables with different instrumental densities.

Part a

1. Provide a rejection algorithm to generate \(\mathcal{N}(0,1)\) random variables, where the instrumental density is Cauchy: \[ g(x) = \frac{1}{\pi (1 + x^2)}, \quad x \in \mathbb{R} \]
   • Determine the efficiency of this algorithm.

Part b

1. Develop an algorithm that generates realizations from a double exponential (Laplace) distribution, with density: \[ g(x) = \frac{\alpha}{2} \exp(-\alpha |x|), \quad \alpha > 0, \; x \in \mathbb{R} \]

2. • Formulate a rejection algorithm to generate \(\mathcal{N}(0,1)\) random variables, where the instrumental density is a double exponential distribution.
   • Find the optimal value of the parameter \(\alpha\) for this density, and determine the corresponding efficiency.

Part c

1. Implement the above algorithms in R, using functions such as rcauchy, runif, rexp, or abs where appropriate.

2. Empirically verify the theoretical efficiencies of each algorithm.

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