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 the expectation and variance of continuous random variables. (Refer to Appendix 1 for more information on th e probability density function, expectation and variance of continuous random variables)

Raffles Institution H2 Mathematics 2025 Year 6 ________________________________________________________________________________________________ ___________________________ Chapter S3: Normal Distribution Page 3 of 26 2 NORMAL DISTRIBUTION AND NORMAL CURVE 2.1 Normal Distribution Consider the following situation: A large number of 1kg bags of sugar are weighed to check how accurately they have been filled, and the results are shown in the histogram below. This histogram is similar to the histograms you might obtain for a variety of different types of data, such as the heights of 5 year-old girls, or the tim e taken to run 2.4km by 18 year-old boys, or the lifespans of batteries, etc, Observe that  the histogram is (almost) symmetrical about the mean, and  the percentages of bags with weight close to th e mean is higher than the percentage of bags with weight further away from the mean. If a smooth curve is drawn through the top of the columns, this will give a distribution which is roughly ‘bell’-shaped. Such a distribution can be modelled by the normal distribution, whose probability density function is given by 21 21f( ) e , 2 x xx       . Note : There is no need to memorize the complicated function above. If a continuous random variable X follows a normal distribution, we write 2~N ( , )X  , where  E( )X  ,  2Var( )X  . Remarks : Recall that in section 1.4 of Chapter S2B, we looke d at the graphs of the probability distribution of the random variable X, where B,Xn p . It can be observed that for large n and value of p which is not too close to 0 or 1, the shape of the graph of the probability distribution of X becomes symmetrical and bell-shaped. In fact, for large n and value of p which is not too close to 0 or 1, the Binomial distribution can be approximated using a normal distribution. (R efer to Appendix 2 for more details) Percentage 1 kg Mass

Raffles Institution H2 Mathematics 2025 Year 6 ________________________________________________________________________________________________ ___________________________ Chapter S3: Normal Distribution Page 4 of 26 y x 2.2 Normal Curve and its Properties Let 2~N ( , )X  . Properties of the normal curve are as follow: (1) It is symmetrical about the line x  . (2) The mean, median and mode are all equal to  . (3) It approaches the x axis as x . (4) Area under the graph gives the probabilities, i.e., P( ) f ( )d b a aXb xx  where f( )yx represents the probability density function of the normal curve. Hence P( )aXb is given by the area under the graph from xa to xb . (5) Total area under the curve is 1. (6) P( ) 0.68 X    P( 2 2 ) 0.95X     P( 3 3 ) 0.997X    i.e., approximately 68%, 95% and 99.7% of the values drawn from a normal distribution lies within 1, 2 and 3 standard deviations of the mean respectively. b a 

Raffles Institution H2 Mathematics 2025 Year 6 ________________________________________________________________________________________________ ___________________________ Chapter S3: Normal Distribution Page 5 of 26  The shape of the normal curve is completely determined by the values of  and  . The following figures illustrate how the mean and the standard deviation affect the shape of the normal curve.  X  Y X Y Figure 1 X and Y have the same standard deviation but different means, with YX Figure 2 X and Y have the same mean but different standard deviations, with XY  (Refer to Appendix 3 for the steps to sketch norm al curves using GC and explore the curves with different means and variances) 2.3 Use of GC to Evaluate Normal Probabilities Let’s revisit our example of 1kg bags of sugar. Suppose the mass of a 1kg bag of sugar follows a normal distribution with mean 1kg and standard deviation 50g. What is the probability that a randomly chosen 1 kg bag of sugar weighs (i) at most 980g? (ii) less than 980g? Let X be the mass of a 1kg bag of sugar in grams. Then 2~ N(1000,50 )X . To evaluate P( 980)X  , we need to find the shaded area under the graph (refer to diagram), i.e. we need to find 980 f( ) dxx  , where f( )yx represents the probability density function of the normal curve. (i) From GC, P( 980) 0.345X  (3 s.f.) (ii) From the sketch, it is clear that P( 980) P( 980)XX = 0.345 (3 s.f.) 0.345 980 1000 X Y

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