The name came to me quickly (truncated normal) but writing out the formal PDF tripped me up a bit.
Start by defining the truncated normal distribution and its parameters, then derive the PDF and CDF using the standard normal PDF and CDF. For sampling, describe at least two methods: inverse transform sampling and rejection sampling, and discuss their trade-offs.
Pro tip: Mention that the truncation point is in standard deviation units if the original distribution is standard normal; otherwise, standardize first. Also, note that the truncated normal is often used in Bayesian inference and survival analysis.
State that the resulting distribution is a truncated normal distribution, specifically left-truncated at 1 (or right-truncated if considering the upper tail).
Write the PDF as f(x) = φ((x-μ)/σ) / (σ * (1 - Φ((a-μ)/σ))) for x > a, where a=1, and the CDF as F(x) = (Φ((x-μ)/σ) - Φ((a-μ)/σ)) / (1 - Φ((a-μ)/σ)) for x ≥ a.
Generate u ~ Uniform(Φ((a-μ)/σ), 1), then set x = μ + σ * Φ^{-1}(u). This is efficient and exact.
Sample from the original normal distribution and reject any sample ≤ a. Repeat until the desired number of samples is obtained. This is simple but may be inefficient if the truncation probability is high.
Compare the methods: inverse transform is exact and efficient but requires the inverse CDF; rejection sampling is easy but can be slow. Mention other methods like Gibbs sampling or specialized algorithms if needed.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.