1. **Problem statement:** How to generate a random variable with a univariate log-concave density. 2. **Definition:** A univariate density $f(x)$ is log-concave if $\log f(x)$ is a concave function. 3. **Common approach:** Use the **Adaptive Rejection Sampling (ARS)** method, which is efficient for log-concave densities. 4. **Key idea of ARS:** Construct an envelope function from tangents to $\log f(x)$ that upper bounds $f(x)$ and sample from this envelope. 5. **Steps in ARS:** - Start with initial points where $\log f(x)$ and its derivative are known. - Build piecewise linear upper bounds (tangents) to $\log f(x)$. - Sample from the exponential of these linear pieces (which form a piecewise exponential distribution). - Accept or reject samples based on the ratio of $f(x)$ to the envelope. 6. **Why ARS works:** The concavity of $\log f(x)$ ensures the envelope is always above $f(x)$, guaranteeing correctness. 7. **Alternative methods:** If ARS is not feasible, consider: - Inverse transform sampling if the CDF is known. - Metropolis-Hastings or other MCMC methods. 8. **Summary:** To generate a random variable with a univariate log-concave density $f(x)$, use Adaptive Rejection Sampling which exploits the concavity of $\log f(x)$ to efficiently sample from $f(x)$.