RI S3 Normal Distribution_Lecture Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 _________________________ Chapter 3: Normal Distribution Page 1 of 26 Chapter S3: Normal Distribution SYLLABUS INCLUDES  Concept of continuous random variables  Concept of a normal distribution as an exampl e of a continuous probability model and its mean and variance; use of 2N( , ) as a probability model  Standard normal distribution  Finding the value of 1P( ) Xx or a related probability, given the values of 1,,x  Symmetry of the normal curve and its properties  Finding a relationship between 1,,x given the value of 1P( ) Xx or a related probability  Solving problems involving the use of E( ) aX b and Var( ) aX b  Solving problems involving the use of E( ) aX bY and Var( ) aX bY , where X and Y are independent PRE-REQUISITES  Concepts of random variable, expectation, variance/standard deviation CONTENT 1 Continuous Random Variable 2 Normal Distribution and Normal Curve 2.1 Normal Distribution 2.2 Normal Curve and its Properties 2.3 Use of GC to Evaluate Normal Probabilities 2.4 Use of GC to Evaluate Inverse Normal Values 3 Standard Normal Distribution 4 Linear Combinations of Independent Normal Random Variables 4.1 Properties of Expectation an d Variance of Random Variables 4.2 Properties of Independent Normal Random Variables 4.3 Random variable 12XX vs random variable 2X

Raffles Institution H2 Mathematics 2025 Year 6 ________________________________________________________________________________________________ ___________________________ Chapter S3: Normal Distribution Page 2 of 26 Appendix 1 Probability Density Function, Exp ectation and Variance of Continuous Random Variables Appendix 2 Approximating a Binomial Dist ribution using a Normal Distribution Appendix 3 Use of GC to sketch Normal Curves Appendix 4 Proof of Mean and Standard Devi ation of Standard Normal Random Variable INTRODUCTION In Chapter S2, we learnt about discrete random variables and a special discrete probability distribution, the binomial distribution. In this chapter, we shall learn about continuous random variables and the most important continuous distribution in statistics – the normal distribution. 1 CONTINUOUS RANDOM VARIABLE Recall that a random variable is a quantity that ta kes different numerical values according to the outcome of a random experiment. A continuous random variable can take any value in a given range, and it best describes data such as height, mass, time, distance, etc. While a discrete random variable is defined by its probability distribu tion, a continuous random variable is defined by its probability density function. The probability density function is represented by a curve f( )yx , and the probabilities are given by the area under the curve. As in the case of discrete random variables, we are also interested in th