H2MA Remedial _ DRV_Solution (RVHS)

Teacher’s copy 1 RVHS H2 Mathematics Remedial Programme Topic: Discrete Random Variables Basic Mastery Questions 1. CJC MYE 9758/2021//Q9 (Parts) A biased red die is such that the probability of any face landing up wards is proportional to the square of the number on that face. The random variable X denotes the score obtained in one throw of this die with ( ) 2P X r kr== , where 1r = , 2, 3, 4, 5, 6, and k is a constant. (i) Find the exact value of k . [2] A second biased die is yellow and the random variable Y denotes the score obtained when the yellow die is thrown once. The probability distribution of Y is y 2 4 6 ( )P Yy= 1 5 2 5 2 5 (ii) Find ( )E Y and show that ( ) 56Var 25Y = . [3] (iii) Given that 1Y and 2Y are two independent observations of Y , find ( )12E YY− and ( )12Var YY− . [2] Answer: (i) 1 91k = (ii) 22 5 (iii) 0, 4.48 (i) r 1 2 3 4 5 6 ( )P Xr= k 4k 9k 16k 25k 36k 4 9 16 25 36 1 91 1 1 91 k k k k k k k k + + + + + = = = (ii) ( ) 1 2 2 22E 2 4 6 5 5 5 5Y      = + + =           ( ) 2 2 2 2 1 2 2 108E 2 4 6 5 5 5 5Y      = + + =          

Teacher’s copy 2 ( ) ( ) ( ) ( ) 22 2 Var =E E 108 22 55 56 shown25 Y Y Y −  =−  = (iii) ( ) ( ) ( )1 2 1 2 22 22E E E 0 55Y Y Y Y− = − = − = ( ) ( ) ( ) ( )1 2 1 2 56 56 112Var Var Var or 4.48 25 25 25Y Y Y Y− = + = + = 2. RI MYE 9758/2021//Q10(i) In a game, a player tosses a fair die, whose faces are numbered from 1 to 6. If the player obtains a 6, he tosses the die a second time, and in this case, his score is the absolute difference of 6 and the second number. Otherwise, his score is the number obtained in the first toss. Let the player’s score be denoted by X. Show that ( ) 7P1 36X == and tabulate the probability distribution of X. [3] ( )P 1 P(first throw = 1) + P(1st throw = 6, 2nd throw = 5) 1 1 1 = 6 6 6 11 6 36 7 (Shown)36 X ==   +     =+ = Probability distribution of X x 0 1 2 3 4 5 P( )Xx= 1 36 7 36 7 36 7 36 7 36 7 36

Teacher’s copy 3 Standard Questions 1. HCI MYE 9758/2020//Q8 (Parts) A bag contains 9 numbered balls of identical size. Four of the balls are numbered 3, three of the balls are numbered 4 and two of the balls are numbered 5. In a game, three balls are drawn from the bag a