LectureNotes9

# LectureNotes9 - Continuous Random Variable and Their...

This preview shows pages 1–7. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

### Page1 / 21

LectureNotes9 - Continuous Random Variable and Their...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online