Analyzing Extreme Tail Probabilities in a Normal Distribution
Consider a standard normal distribution \(Z \sim N(0,1)\). For large values of \(z\), the upper tail probability is often approximated by the formula: \( P(Z \geq z) \approx \frac{1}{z\sqrt{2\pi}} e^{-\frac{z^2}{2}} \). (a) Explain why the exponential term \(e^{-\frac{z^2}{2}}\) is the dominant factor in determining the tail probability for large \(z\). (b) Using this approximation, estimate […]