Normal Distribution and Standard Normal Curve

Lesson 30 of STA301 explores the prominent continuous probability model: the Normal (Gaussian) Distribution, introduced by Carl Friedrich Gauss. The normal distribution underpins statistical inference, modeling natural phenomena, measurement errors, and sampling distributions. The mathematical equation of the normal probability density function is f(x) = [1 / (sigma * sqrt(2*pi))] * exp[- (x - mu)^2 / (2*sigma^2)] for -inf < x < +inf, governed by two parameters: the population mean mu (-inf < mu < +inf) determining location, and standard deviation sigma (sigma > 0) determining spread. The lecture details the mathematical properties of the normal curve: it is unimodal, bell-shaped, and perfectly symmetrical around x = mu; the mean, median, and mode are identical (mu = Median = Mode); the curve is asymptotic to the horizontal axis; total area under the curve equals unity (1.0); points of inflection occur at x = mu +- sigma; odd central moments are zero due to symmetry (mu_1 = mu_3 = 0); and its kurtosis ratio beta2 equals exactly 3 (mesokurtic). To compute probabilities without integrating the non-elementary Gaussian function, any normal variable X is converted into the Standard Normal Variable Z via the standardizing transformation: Z = (X - mu) / sigma. The standard normal distribution has mean 0 and variance 1, denoted Z ~ N(0, 1). Students learn to read Standard Normal Probability Tables to find cumulative areas Phi(z) = P(Z <= z), utilizing symmetry relations such as P(Z >= z) = 1 - Phi(z) and P(Z <= -z) = 1 - Phi(z). The lecture also explores the Normal Approximation to the Binomial Distribution: when n is large and neither p nor q is close to zero (np >= 5 and nq >= 5), the discrete binomial variable is approximated by normal distribution N(np, npq). To reconcile discrete integers with continuous intervals, the Continuity Correction Factor (+- 0.5) is applied: P(X = x) becomes P(x - 0.5 <= X_cont <= x + 0.5). In Virtual University final term examinations, finding Z-score probabilities, applying continuity corrections, and inverse normal calculations represent core computational questions. Reviewing sta301 handouts and solved past papers guarantees academic success.