RI S2A Discrete Random Variables_Tutorial Qns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 __________________________________ Tutorial S2A: Discrete Random Variables Page 1 of 4 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 Xx 0.04 0.16 0.24 0.16 0.15 0.1 0.1 0.05 [ –0.83, 1.86] 2 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 BA B X A BA 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 ] 3 A discrete random variable X takes values 2, 3, 4, 5 with probabilities as shown in the table. x 2 3 4 5 P( )Xx k 4 k 9 k 16 k (i) Find k, leaving your answer as a fraction. [2] (ii) Find E( cos )X , giving your answer to 3 significant figures. [3] (iii) Find 12P( 7)XX , where 1X and 2X are two independent observations of X. [3] [(i) 144 205k  (ii) 0.530 (iii) 0.0891]

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ __________________________________ Tutorial S2A: Discrete Random Variables Page 2 of 4 4 A box contains 2 fair tetrahedral dice. The firs t die has sides labeled 1, 1, 2, and 3 and the second die has sides labeled 1, 2, 3 and 3. A die is taken at random from the box and thrown. X, the score is defined as follows: If the first die is picked, then it is thrown and the score is defined as two times the number which appears on the base of the first die. If the second die is picked, then it is thrown and the score is the number which appears on the base of the second die. Show that P(X = 2) = 3 8 , and find the probability distribution of X. (i) Show that E(X) = 23 8 and find the exact value of Var(X). (ii) If this experiment is performed twice, find the probability that the score is 3 for the first experiment given that the total score is 4. (iii) Tom and Jerry take turns to perform this expe riment with Jerry playing first. They stop when one of them obtains a score of 2. What is the probability of Tom obtaining a score of 2? [(i) 135 64 (ii) 2 13 (iii) 5 13 ] 5 Alfred and Bertie play a game, each starting with cash amounting to £100. Two dice are thrown. If the total score is 5 or more then Alfred pays £x, where 08 x , to Bertie. If the total score is 4 or less, then Bertie pays £(x + 8) to Alfred. By finding the probability distribution of Y, where Y denotes the random variable representing Alfred’s gains after one game, show that the expectation of Alfred’s cash after the first ga