Define and sample a truncated normal

Define the truncated normal variable $Z \mid a < Z < b$ for $Z \sim N(0,1)$: write down its normalized pdf and cdf. Next, construct efficient samplers for three cases: (i) $a = 1, b = +\infty$; (ii) $a = 4, b = 4.05$; (iii) $a = 4, b = +\infty$. For each case, contrast inverse-CDF sampling — draw $U \sim \mathrm{Uniform}(\Phi(a), \Phi(b))$ and set $X = \Phi^{-1}(U)$ — with simple rejection sampling from $N(0,1)$ by computing the expected acceptance rate: $1-\Phi(1)$, $\Phi(4.05)-\Phi(4)$, and $1-\Phi(4)$. Suggest an algorithm that works well in extreme tails (for example, exponential or half-normal proposals within acceptance–rejection), discuss the numerical stability of evaluating $\Phi$ and $\Phi^{-1}$ near machine precision, and describe how you would validate the sampler (a KS test on back-transformed uniforms, plus moment checks).

Overview: This question tests skill with probability distributions and Monte Carlo sampling methods in the Statistics & Math area, centered on the truncated normal distribution, its normalized pdf/cdf, and the design and comparison of samplers for moderate and extreme tail regimes.