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