RI S2A Discrete Radnom Varible Lecture Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 ___________________________________ Chapter S2A: Discrete Random Variables Page 1 of 21 Chapter S2A: Discrete Random Variables SYLLABUS INCLUDES  Concepts of discrete random variables, probability distributions, expectations and variances PRE-REQUISITES  Probability  Descriptive Statistics (See Appendix) CONTENT 1 Random Variables 1.1 Definition 1.2 Discrete versus Cont inuous Random Variables 1.3 Probability Distribution of a Discrete Random Variable 1.4 Expectation of a Di screte Random Variable 1.5 Relationship between E xpectation and Sample Mean 1.6 Independent Random Variables 1.7 Functions of a Discrete Random Variable 1.8 Variance and Standard Deviatio n of a Discrete Random Variable 1.9 Linear Combination of Independent Random Variables 1.10 Miscellaneous examples Appendix: A A Simple Application of Statistics in Insurance B Descriptive Statistics (Assumed knowledge) INTRODUCTION It often happens in probability that the events we are interested in involve counting or measuring something. When we toss two dice, what are the possible outcomes we can measure?  Number of ‘1’s obtained  Number of even outcomes  Sum of two dice, etc.

Raffles Institution H2 Mathematics 2025 Year 6 ___________________________________________________________________________________________ ___________________________________ Chapter S2A: Discrete Random Variables Page 2 of 21 For instance, when tossing a pair of dice, we ar e often interested in the sum of the two dice and not overly concerned over the value of each individual die. That is, we may be only interested in knowing that the sum is 7 and not whether the actual outcome is {1,6}, {2,5}, {3,4}, {4,3}, {5,2} or {6,1}. Also in coin-flipping, we may be only interested in the number of heads obtained and not be too concerned about the actual head-tail sequences that result. These quantities of interests, or more formally, these real value functions defined on the sample space, are known as random variables. 1 RANDOM VARIABLE 1.1 Definition A random variable is a quantity that takes different numerical values according to the result of a random experiment whose outcome cannot be predicted exactly. For example, consider the experiment of tossing a fair coin 3 times and observe the outcome. The sample space  may be represented by {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Let’s denote the number of heads obtained when tossing the fair coin 3 times by X.  The possible values X can take are 0, 1, 2 and 3 (variable).  The value it assumes is subject to chance (random), so X is a random variable.  In general, P (X = x) refers to the probability of the random variable X assuming a specific value x.  Eg. P( 2)X  refers to the probability of obtaining 2 heads in the 3 tosses. 1.2 Discrete versus Continuous Random Variables A discrete random variable takes on a countable 1 number of possible values while a continuous random variable can take any value in a given range. Which of the following are discrete random variables and which are continuous?  The number of phone calls a person picks up in a minute  The life span of a light-bulb manufactured by a factory  The number of students who are late for school on a particular day  The distance travelled by a car in a week  The height of a student from the Class of 2020 1 A countable set is one whose elements are in 1-1 correspondence with the positive integers. Important! We use capital letters to denote random variables, eg. X, Y, and lower case letters for the values they take, eg. x and y. Avoid using Z, B, P and N.

Raffles Institution H2 Mathematics 2025 Year 6 ___________________________________________________________________________________________ ___________________________________ Chapter S2A: Discrete Random Variables Page 3 of 21 In general, random variables that represent the number of occurrences of an event or objects are most suited for discrete random variables whil e those that measure spatial dimensions, time, temperature, weight etc, are best described by continuous random variables. Very often, random variables are associated with their probability distributions ( for discrete random variables) or probability density functions (for continuous random variables), as well as expectation and variance. In this chapter, we will focus on discrete random variables and their associated probability distributions, expectation and variance. 1.3 Probability Distribution of a Discrete Random Variable For a random experiment, Let S be the sample space of the experiment (e.g. outcomes of tossing 3 coins). Let X be the random variable defined to take valu es of the experiments (e.g. number of heads). A table or formula giving the values of P( )X x for every x in S is called the probability distribution of X. For the experiment of tossing a fair coin 3 times where X is the number of heads obtained, the probability distribution of X is as follows x 0 1 2 3 P( )X x 1 8 3 8 3 8 1 8 Observe that the probability distribution of X satisfy the following: 1. 0P ( )1Xx  for all x in S. 2. P( ) 1 xS Xx   where the summation is over all values of x in S. 1.4 Expectation of a Discrete Random Variable The expectation (or mean, or expected value) of a discrete random variable X is the weighted average of the possible values that X can take on, each value being weighted by its corresponding probability. It is denoted by E(X) or  and is a measure of the ‘centre’ of the probability distribution of X. Formally, if X is a discrete random variable taking values from a set S, then E( ) P( ) . xS Xx X x  

Raffles Institution H2 Mathematics 2025 Year 6 ___________________________________________________________________________________________ ___________________________________ Chapter S2A: Discrete Random Variables Page 4 of 21 Example 1 Let X be the random variable representing the sum of scores obtained when 2 fair dice are thrown. State the smallest set S that contains all the possible values of X. Tabulate the probability distribution of X. Find E(X). Solution: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 Since each of the 36 possibilities are equally likely to occur, they each carry a probability of 1 36 . Therefore, the probability distribution of X is x 2 3 4 5 6 7 8 9 10 11 12 P( )Xx 1 36 21 36 18 31 36 12 41 36 9 5 36 61 36 6 5 36 41 36 9 31 36 12 21 36 18 1 36 12 2 Thus, E( ) P( ) 7 x Xx X x    d1 d2 d1+d2

Raffles Institution H2 Mathematics 2025 Year 6 ___________________________________________________________________________________________ ___________________________________ Chapter S2A: Discrete Random Variables Page 5 of 21 Example 2 The probability distribution of a discrete random variable is given by 2 1 , i f 0,10P( ) , if 1, 2,3, 4. y Yy cy y      Given that c is a constant, find the value of c. With this value of c, find E( )Y . Solution: As 4 0 P( ) 1 y Yy   1 491 6110 930 10 9 300 3 100 ccc c c c c       4 0 E( ) P( ) y Yy Y y   4 3 1 130 10 100y y         = 3 1.5 Relationship be tween Expectation and Sample Mean Consider the number of heads obtained when we toss 3 fair coins. Denoting the number of heads obtained by X, the possible values X can take are 0, 1, 2 or 3. By working out the values of P( X = 0), P( X = 1), P( X = 2) and P( X = 3), the probability distribution of