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Unformatted text preview: Continuous Random Variable and Their Probability Distributions: Part III Cyr Emile M’LAN, Ph.D. [email protected] Continuous Random Variables: Part III – p. 1/21 Introduction ♠ Text Reference : Introduction to Probability and Its Application, Chapter 5. ♠ Reading Assignment : Sections 5.6, March 30  April 1 In this set of notes, we will discuss the most important and most widely used distribution in statistical application. In fact, many numerical measurements (heights, weights, reaction time in psychological experiments, intelligence and aptitude scores) have distributions that can be approximated by a normal curve. In addition, whenever an average of random variables is computed, it tends to be spread out like the variation in a normal curve. Continuous Random Variables: Part III – p. 2/21 The Standard Normal Distribution Density and Distribution Function Let Z be a continuous random variable with density f ( z ) = 1 √ 2 π e z 2 2 ,∞ < z < ∞ . Then Z has a standard normal distribution , denoted by Z ∼ N (0 , 1) . Sometimes, we also use φ ( z ) to denote the probability density function of Z , i.e., φ ( z ) = 1 √ 2 π e z 2 2 ,∞ < z < ∞ . Continuous Random Variables: Part III – p. 3/21 The Standard Normal Distribution The corresponding cumulative density function of Z is given by Φ( z ) = P ( Z ≤ z ) = integraldisplay z∞ φ ( u ) du = integraldisplay z∞ 1 √ 2 π e u 2 2 du. The standard normal distribution is symmetric about z = 0 , i.e., φ ( z ) = φ ( z ) and Φ( z ) = 1 Φ( z ) . Φ( z ) is tabulated on page 448 of textbook. Continuous Random Variables: Part III – p. 4/21 The Standard Normal Distribution Mean and Variance We have E [ Z ] = 0 Var ( Z ) = 1 Example 5.13 : If Z ∼ N (0 , 1) , then we have the following 3 σ rule: P ( 1 ≤ Z ≤ 1) ≈ . 68 = 68% , P ( 2 ≤ Z ≤ 2) ≈ . 95 = 95% , P ( 3 ≤ Z ≤ 3) ≈ . 997 = 99 . 7% . Continuous Random Variables: Part III – p. 5/21 The General Normal Distribution Density A random variable X is said to be normal , if its density is given by f ( x ) = 1 √ 2 πσ e ( x μ ) 2 2 σ 2 ,∞ < x < ∞ , where∞ < μ < ∞ is a location parameter and σ > is a spread parameter....
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 Spring '09
 Normal Distribution, Probability, Probability distribution, Probability theory, probability density function, Cumulative distribution function

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