ACJC 2026 DRV and Binomial Distributions Lecture Notes

1 14 DISCRETE RANDOM VARIABLES AND BINOMIAL DISTRIBUTION SYLLABUS • Concept of discrete random variables, probability distributions, expectations and variances. • Concept of the binomial distribution B( , )np as an example of a discrete probability distribution. • Use of B( , )np as a probability model, including conditions under which the binomial distribution is a suitable model. • Use of the mean and variance of the binomial distribution.

ACJC 2025/26 H2 Mathematics (9758) 2 CONTENTS 1 Definitions ........................................................................................ 3 2 Probability Distribution Function ..................................................... 4 3 Cumulative Distribution Function .................................................... 6 4 Expectation ....................................................................................... 9 4.1 Expectation of a Function of a Discrete Random Variable .... 9 4.2 Properties of Expectation ........................................................ 9 5 Variance .......................................................................................... 11 5.1 Properties of Variance ........................................................... 12 6 Miscellaneous Examples ................................................................ 16 7 Binomial Distribution ..................................................................... 19 7.1 Definition of a Binomial Distribution ................................... 20 7.2 Examples of Binomial Distribution with Known Parameters ............................................................................................... 21 7.3 Examples of Binomial Distribution with Unknown Parameters ............................................................................................... 26 7.4 Self Exploration on the Binomial Distribution ..................... 32 Annex A: Practice Questions on Discrete Random Variables and Binomial Distributions .......................................................... 33

14 DRV and Binomial Distribution 3 LECTURE 1 Lesson Outline • Introduction to discrete random variables: definition, probability distribution function, cumulative distribution function and expectation. 1 DEFINITIONS A random variable is a function that associates every outcome of an experiment with a unique numerical value. For example, consider the experiment of tossing a fair die. The number that shows up on the die roll can be denoted by a random variable X, where the value of X may correspond to the six outcomes of the die roll: 1, 2, …, 6. A discrete random variable is a random variable which may only take on a countable number of distinct numerical values such as 0, 1, 2, …. The random variable X in the example above is a discrete random variable. A continuous random variable is a random variable which can take on an uncountable number of numerical values (i.e. a range of values). For example, if the random variable Y denotes any real number between 0 and 1, then Y is a continuous random variable. Example 1 (Concept of Random Variables) X denotes a random variable. Write down the possible values of X and conclude whether it is a discrete or continuous random variable. Random Variable X Possible Values of X Discrete or Continuous The number of students s itting next to a chosen student. The number of pens in a student’s pencil case. The number of Instagram followers that a student currently has. The amount of time a student spent on homework yesterday. The height of a student.

ACJC 2025/26 H2 Mathematics (9758) 4 Notes • We use upper case alphabet letters to represent random variables. We use lower case alphabet letters to represent the values these random variables can take. • The expression P( )Xx= denotes the probability of the random variable X taking on the value x. Example 2 Let the random variable X denote ‘the number of fours obtained when two dice are thrown’. Find (i) P( 0)X = (ii) P( 1)X = (iii) P( 2)X = Solution (i) P( 0)X == 2 5 25 6 36  = (ii) P( 1)X == 1 5 52 6 6 18    =     (iii) P( 2)X == 2 11 6 36  = ■ 2 PROBABILITY DISTRIBUTION FUNCTION Let X be a discrete random variable. The probability distribution function (pdf) (or probability function) of X is defined by P( )Xx= , where x denotes all the possible values that the random variable X can take. The pdf of X may be presented in a table form or in a formula that lists all the probabilities of the possible values of X. In this case, we may consider the table as the probability distribution of X. For example, let the random variable X denote the number of heads obtained when tossing a fair coin once. Then the probability distribution of X can be represented by either of the following: x 0 1 P( )Xx= 1 2 1 2 or 1 , 0 or 1P( ) 2 0, otherwise xXx  ===   This probability distribution has a name - the Bernoulli distribution.

14 DRV and Binomial Distribution 5 Properties of the probability distribution p(x) of a discrete random variable X 1. For all values of x, 0 p( ) 1x . 2. all p( ) 1 x x = . That is, if the variable X can take on values 12, , ..., nx x x such that 11P( )X x p== , 22P( )X x p== and so on, then 12 ... 1 np p p+ + + = . Example 3 Two fair tetrahedral dice, each with faces labeled 1, 2, 3 and 4, are thrown. If X is the random variable denoting ‘the sum of the outcomes on the bottom faces of the two tetrahedral dice’, find the probability distribution of X. Solution 1 2 3 4 1 2 3 4 P( ) P( )XX= = = = P( ) P( )XX= = = = P( ) P( )XX= = = = P( )X == ■

ACJC 2025/26 H2 Mathematics (9758) 6 Example 4 The probability function of a discrete random variable Y is given by 2P( )Y y cy== , for 0,1, 2, 3, 4.=y Find the value of the constant c. Solution 4 0 P( ) 1 y Yy = == 4 2 0 2 2 2 2 2 1 (0 1 2 3 4 ) 1 130 1 30 y cy c cc = = + + + + = =  =  3 CUMULATIVE DISTRIBUTION FUNCTION (NOT IN SYLLABUS) For a discrete random variable X, the cumulative probabilities are obtained by summing all the probabilities up to a particular value. If X is a discrete random variable with probability function P( )Xx= for 12, , ..., nx x x x= , then the cumulative distribution function of X is given by F( )t , where F( ) P( ) P( ) , where xt t X t X x t  =  = =   Note • ( ) ( )P 1 PX a X a = −  . 4 EXPECTATION The expectation of X (or the mean/expected value of X), written as E( ),X is defined as all E( ) P( ) x X x X x== . Notes 1. E( )X is a constant. 2. E( )X represents the mean outcome if the same experiment is repeated under the same conditions a large number of times.

14 DRV and Binomial Distribution 7 Example 5 (Frequency Table) The number of books borrowed by 120 children in a month are shown below. Number of books, x 1 2 3 4 5 6 Frequency (number of children), f( )x 15 22 23 19 23 18 Probability that a child has borrowed x books, P( )Xx= 15 120 22 120 23 120 19 120 23 120 18 120 Find the mean number of books borrowed by a child in a month. Solution Method 1: ‘O’ Level method Mean number of books total number of books borrowed by the children total number of children= all all f( ) f ( ) 1(15) 2(22) 3(23) 4(19) 5(23) 6(18) 15 22 23 19 23 18 3.56 (3 s.f.) x x xx x= + + + + += + + + + + =   Method 2: Using expectation formula Mean number of books all E( ) P( ) 15 22 23 19 23 181 2 3 4 5 6120 120 120 120 120 120 3.56 (3 s.f.) x X x X x = ==            = + + + + +                       =  So, the mean number of books borrowed is 3.56. ■ Connecting what we learnt in secondary school and the expectation formula, f( ) f( ) P