Normal Distribution: Z-Scores, Standard Normal, and CLT for Exam P

The Normal Distribution

If X ~ Normal(mu, sigma^2), its PDF is f(x) = (1 / (sigma * sqrt(2*pi))) * exp(-(x - mu)^2 / (2*sigma^2)). The distribution is symmetric about mu, with mean = median = mode = mu and variance sigma^2. About 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

The standard normal Z = (X - mu) / sigma has mean 0 and variance 1. All normal probability calculations reduce to looking up values in the standard normal table (or using the function Phi). P(X <= x) = Phi((x - mu) / sigma).

Z-Score Calculations

For Exam P, you must be comfortable converting between X-values and Z-scores. If claim amounts are Normal(500, 100^2), the probability that a claim exceeds 650 is P(X > 650) = P(Z > 1.5) = 1 - Phi(1.5) = 1 - 0.9332 = 0.0668.

To find a percentile, invert: the 90th percentile of X is mu + z_{0.90} * sigma = 500 + 1.2816 * 100 = 628.16. Exam P provides a normal table, so memorize key values: z_{0.90} = 1.282, z_{0.95} = 1.645, z_{0.975} = 1.960, z_{0.99} = 2.326.

Linear Combinations

If X1, X2, ..., Xn are independent normal random variables, any linear combination a1*X1 + a2*X2 + ... + an*Xn is also normally distributed with mean = sum(ai*mu_i) and variance = sum(ai^2 * sigma_i^2). This property is unique to the normal distribution and simplifies many Exam P problems involving sums or differences of normal variables.