ACJC 2026 Normal Distribution Lecture Notes

1 15 NORMAL DISTRIBUTION SYLLABUS • Concept of a normal distribution as an example of a continuous probability model and its mean and variance; use of 2N( , )µσ as a probability model • Standard normal distribution • Finding the value of P( )Xx< or a related probability, given the values of x, µ, σ • Symmetry of the normal curve and its properties • Finding a relationship between x, µ, σ given the value of P( )Xx< or a related probability • Solving problems involving normal random variables • 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 Exclude normal approximation to binomial distribution

ACJC 2025/26 H2 Mathematics (9758) 2 CONTENTS 1 Continuous Random Variables ......................................................... 3 1.1 Probability Density Function .................................................. 3 1.2 Expectation of a Continuous Random Variable, E( )X .......... 4 1.3 Variance of a Continuous Random Variable, Var( )X ........... 4 2 Normal Distribution.......................................................................... 5 2.1 Properties of the Probability Density Function of a Normal Random Variable .................................................................... 5 2.2 Effect of µ and σ on Probability Density Curve of a Normal Distribution ............................................................................. 7 2.3 Finding Probabilities Using the Graphing Calculator ............. 7 2.4 Finding the Inverse Normal .................................................. 10 2.5 Standardising a Normal Distribution .................................... 13 3 Linear Combination of Independent Normal Random Variables... 15 Annex A: Approximation of the Binomial Distribution by a Normal Distribution ........................................................................... 22 Annex B: Practice Questions on Normal Distribution .......................... 23

15 Normal Distribution 3 LECTURE 1 Lesson Outline • Introduction to continuous random variables: definition, probability density function, expectation and variance. • Understand the properties of the probability density function of a normal distribution and how the parameters of the normal distribution affect its curve. • Understand and calculate probabilities involving normal distributions 1 CONTINUOUS RANDOM VARIABLES A continuous random variable (CRV) has random values which are defined over a range, i.e., the random values are not countable. Examples include the length of time a person waits for a taxi and the volume of water consumed by a household in a day. 1.1 Probability Density Function A function f( )x is the probability density function (pdf) of a continuous random variable X if (i) f( ) 0x ≥ for all real values of x, (ii) P( ) f ( ) d b a aXb xx≤≤= ∫ , (iii) f( ) d 1xx ∞ −∞ =∫ . Notes • P( ) f( ) d 0 a a Xa xx= = =∫ • P( ) P( ) P( ) P( )aXb aXb aXb aXb≤≤= ≤<= <≤= << ( P( ) P( ) 0Xa Xb= = = = ) • P( ) 1 P( )Xa Xa>= − ≤ • P( ) P( ) P( )aXb Xb Xa≤≤= ≤− < This is the shaded area. x a b

ACJC 2025/26 H2 Mathematics (9758) 4 1.2 Expectation of a Continuous Random Variable, E(X) Recall from Chapter 14 that for a discrete random variable X , its expectation E( )X is given by all E( ) P( ) x X xXx= =∑ . For a continuous random variable X, the expectation of X (expected value or mean), E( )X , is given by E( ) f( ) dX xx x ∞ −∞ = ∫ . Notes • E( )X is a constant. • E( )X represents the expected outcome if a sufficiently large number of the same experiment is repeated under the same conditions. 1.3 Variance of a Continuous Random Variable, Var(X) The most common measure of spread of data is the variance (or standard deviation, where standard deviation = variance ). Like the expectation, variance makes use of the information contained in all of the observations. The variance is a measure of the spread of the observations from the mean. It is actually a mean of the squared deviations of the observations from the mean. Consider the continuous random variable X and let E( )X µ= . The variance of X, written Var( )X , of a random variable X is given by ( ) 2Var( ) E ( )XX µ= − . The standard deviation of X is Var( )Xσ = . The variance and the standard deviation are both at least zero. The greater the deviations, the greater the variance (and the standard deviation) will be.

15 Normal Distribution 5 2 NORMAL DISTRIBUTION The normal distribution is the most important distribution in statistical theory because it is a suitable model for a very large number of distributions of data. A continuous random variable X is said to have a normal distribution with mean µ and variance 2σ if X has probability density function 2 2 () 21f( ) e 2 x x µ σ σ −− = π , where x−∞ < < ∞. (You are not required to remember the formula.) If X has a normal distribution with mean µ and variance 2σ , we write it as ( ) 2~N ,X µσ . 2.1 Properties of the Probability Density Function of a Normal Random Variable The probability density function of a normal random variable is a bell - shaped curve that has the following properties: (i) It is symmetrical about its mean µ. (ii) The mean, mode and median coincide at x µ= . (iii) There is only one peak at x µ= . (iv) The curve which lies completely above the x-axis (since f( ) 0x > for all real values x) and tapers off rapidly at each “tail” i.e. f( ) 0x +→ as x → ±∞. (v) Total area under the curve f( )d 1xx ∞ −∞ = =∫ . x µ µ − σ µ + σ

ACJC 2025/26 H2 Mathematics (9758) 6 As the probability is the area under the curve, therefore P( ) P( ) P( ) P( ) f ( ) d b a aXb aXb aXb aXb xx≤≤= ≤<= <≤= <<= ∫ , where f( )x is the probability density function of X in the normal distribution. Notes • Approximately 68.3% of the data falls within one standard deviation about the mean. • Approximately 95.4% of the data falls within two standard deviations about the mean. • Approximately 99.7% of the data falls within three standard deviations about the mean. x a b x 68.3% 95.5% 99.7%

15 Normal Distribution 7 2.2 Effect of μ and σ on Probability Density Function of a Normal Distribution Density Curves of Normal distributions with the same mean but different variances where 222 123σσσ<< Density Curves of Normal distributions with same variance but different means where 123µµµ<< Normal Distribution An app showing the effect of changing the mean and standard deviation on the normal distribution curve. https://www.geogebra.org/m/E7R55dsW 2.3 Finding Probabilities Using the Graphing Calculator The function normalcdf( in the TI GC can be used to calculate P( )aXb<< , where a is the lower bound and b is the upper bound. If the values µ and σ are not specified, the default values are 0 and 1 respectively. x µ , smallest variance , largest variance x , largest mean , smallest mean

ACJC 2025/26 H2 Mathematics (9758) 8 Example 1 It is given that 2~ N(10, 4 )X . For each of the following probabilities, (a) P(8 12)X<≤ (b) P( 14.8)X ≤ (c) P( 6)X > (i) sketch the curve of the distribution of X and shade the region representing the probability, (ii) find the probability, giving your answer correct to 3 decimal places. Solution (a)(i) P(8 12)X<≤ 1) Press [2nd] [VARS] and select 2:normalcdf(. 2) Enter the values of a, b, µ and σ. 3) Select Paste into the main screen and press [ENTER] (ii) P(8 12)X<≤ = (b)(i) P( 14.8)X ≤ 1) Follow the same three steps as in (a) except that the lower bound is −∞ . This is represented by 9910− in the calculator. 2) It is keyed in using the keystrokes [(-)] [