RI S2A Discrete Random Variables_Tutorial Solns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 __________________________________ Tutorial S2A: Discrete Random Variables Page 1 of 15 Tutorial S2A: Discrete Random Variables Section A (Discussion Questions) 1 Find the mean and standard deviation of the random variable X with the following probability distribution: x 4 3 2 1 0 1 2 3 P X x 0.04 0.16 0.24 0.16 0.15 0.1 0.1 0.05 [ –0.83, 1.86] E( ) P( )X x X x  =                  4 0.04 3 0.16 2 0.24 1 0.16 0 1 0.1 2 0.1 3 0.05           = –0.83 2 2E( ) P( )X x X x  = 4.15 2 2 Var( ) E( ) E( )X X X  =  24.15 0.83  = 3.4611   Standard deviation = 1.86 3 s.f.  Using GC, Enter data into 2 lists Read off the values from “sample mean”, x and “sample standard deviation”, x . E(X) = -0.83 SD(X) = 1.86 (3sf) Estimated number of tutorials: 2

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ __________________________________ Tutorial S2A: Discrete Random Variables Page 2 of 15 2 9233/1980/02/Q11 (modified) Two dice are thrown and the numbers A and B shown on each die are noted. The score X from the throw is defined by: if , if . A B A BX A B A B     (i) Tabulate the probability distribution of X. (ii) Evaluate the exact value of E(X) and Var(X). [(i) 28 9 (ii) 1015 162] 1 2 3 4 5 6 1 2 1 2 3 4 5 2 1 4 1 2 3 4 3 2 1 6 1 2 3 4 3 2 1 8 1 2 5 4 3 2 1 10 1 6 5 4 3 2 1 12 Probability distribution of X: x 1 2 3 4 5 6 8 10 12 P X x 10 5 36 18 9 1 36 4 6 1 36 6 5 36 2 1 36 18 1 36 1 36 1 36 1 36 E( ) P( )X x X x            10 9 6 5 2 1 11 2 3 4 5 6 836 36 36 36 36 36 36 1 1 10 1236 36                                                          = 28 9 2 2E( ) P( )X x X x            2 2 2 2 2 2 2 2 2 10 9 6 5 2 1 11 2 3 4 5 6 836 36 36 36 36 36 36 1 1 10 1236 36 287 18                                                            d1 d2 x

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ __________________________________ Tutorial S2A: Discrete Random Variables Page 3 of 15 2 2V ar( ) E( ) E( )X X X  2287 28 18 9       = 1015 162 The following GC screenshots are for checking purpose only (note that question asks for exact values of E(X) and Var(X)) 3 MJC Prelim 9233/2005/02/Q12 A discrete random variable X