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The Normal Distribution The normal or bell shaped distribution is the most widely used distribution. Definition: A random variable 𝑌 has a normal distribution if and only if 𝑓(𝑦) = 1 2 /(2𝜎 2 ) 𝜎√2𝜋 𝑒 −(𝑦−𝜇) −∞ < 𝑦 < ∞ and − ∞ < 𝜇 < ∞, 𝜎 > 0. Theorem: If 𝑌 has a normal distribution, then 𝐸(𝑌) = 𝜇 and 𝑉(𝑌) = 𝜎 2 . If 𝑌 has a normal distribution with mean 0 and standard deviation 1, then 𝑓 (𝑦) = 1 √2𝜋 𝑒 −𝑦 2 /2 −∞ < 𝑦 < ∞ In this case, 𝑌 is a standard normal distribution and is denoted by 𝑍. The probability of a normal distribution is on page 848. 1 2 Example 4.8 Let Z denote a normal random variable with mean 0 and standard deviation 1. a. Find 𝑃(𝑍 > 2) (Answer is 0.0228) b. Find 𝑃(−2 ≤ 𝑍 ≤ 2) (Answer is 0.9544) c. Find 𝑃(0 ≤ 𝑍 ≤ 1.73) (Answer is 0.4582) Example 4.9 The achievement scores for a college entrance examination are normally distributed with mean 75 and standard deviation 10. What fraction of the scores lies between 80 and 90? 𝑦1 − 𝜇 80 − 75 𝑧1 = = = 0.5 𝜎 10 𝑦2 − 𝜇 90 − 75 𝑧2 = = = 1.5 𝜎 10 𝐴 = 𝐴0.5 − 𝐴1.5 = 0.3085 − 0.0668 = 0.2417. 3