Normal distribution University of Notre Dame（正态分布圣母大学）.pdf

Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a standard of reference for many probability problems. I. Characteristics of the Normal distribution • Symmetric, bell shaped • Continuous for all values of X between -∞ and ∞ so that each conceivable interval of real numbers has a probability other than zero. • -∞≤ X ≤∞ • Two parameters, µ and σ. Note that the normal distribution is actually a family of distributions, since µ and σ determine the shape of the distribution. • The rule for a normal density function is 1 2 2 f(x; µ,σ2 ) = e-(x-µ) /2σ 2πσ2 • The notation N(µ, σ2 2 ) means normally distributed with mean µ and variance σ . If we say 2 2 X ∼ N(µ, σ) we mean that X is distributed N(µ, σ). • About 2/3 of all cases fall within one standard deviation of the mean, that is P(µ - σ≤ X ≤ µ + σ) = .6826. • About 95% of cases lie within 2 standard deviations of the mean, that is P(µ - 2σ≤ X ≤ µ + 2σ) = .9544 Normal distribution - Page 1 II. Why is the normal distribution useful? • Many things actually are normally distributed, or very close to it. For example, height and intelligence ar