2023 Stats3 Discrete Random Variables APQ solutions (RVHS)

Statistics 3 Tutorial: Discrete Random Variables Additional Practice Questions 1. An unbiased cubical dice has three faces numbered ‘1’, two faces numbered ‘2’ and one face numbered ‘3’. The random variable X is the number showing on the top face of the dice when it is thrown. Show that 5E( ) 3X = and find Var( )X . Solution: X = x 1 2 3 P(X = x) 3 6 2 6 1 6 2 2 2 2 2 3 4 3 10 5E( ) 6 6 6 6 3 3 2 1 20E( ) 1 2 3 6 6 6 6 20 5 5Var( ) 6 3 9 X X X = + + = = = + + = = − = 2. [2001/II/6] A random variable X has the probability distribution given in the following table. x 2 3 4 5 P(X = x) p 2 10 3 10 q (i) Given that E(X) = 4, find p and q. (ii) Show that Var(X) = 1. (iii) Find E(|X – 4|). (iv) Ten independent observations of X are taken. Find the probability that the value 3 is obtained at most three times. Solutions: (i) 23E( ) 4 2 3 4 5 4 10 10 11 2 5 (1) 5 X p q pq    =  + + + =        + = −−− Also, 2 3 1 1 (2)10 10 2p q q p+ + + =  = − −−− Subs. (2) into (1): 1 1125 25pp + − = 12 10 5pq=  =

(ii) 2 2 2 2 2 1 2 3 2E( ) 2 3 4 5 1710 10 10 5X        = + + + =               2 2 2Var( ) E( ) E( ) 17 4 1X X X= − = − = (shown) (iii) 1 2 3 2 4E(| 4 |) | 2 | | 1| | 0 | |1|10 10 10 5 5X        − = − + − + + =               (iv) Let Y denote the no. of times (out of 10) ‘3’ is obtained. Then 2~ B 10, 5Y   Required probability = ( 3) 0.879PY = (to 3 sf) 3. [2017 PJC MYE/ P2/ Q7] In a game, 3 red balls and 7 white balls are placed in a bag. A player draws 3 balls at random and without replacement. The player scores 2 poi nts for every red b all that is drawn, and 1 point for every white ball that is drawn. The total score X is obtained by adding up the scores of the 3 balls. Find the probability distribution of X. [3] Find ( )E X and show that ( )Var 0.49X = . [2] x 3 (3W) 4 (2W1R) 5 (1W2R) 6 (3R) ( )P Xx= 7 6 5 7 10 9 8 24  = OR 7 3 10 3 7 24 C C = 7 6 3 3! 21 10 9 8 2! 40   = OR 73 21 10 3 21 40 CC C = 7 3 2 3! 7 10 9 8 2! 40   = OR 73 12 10 3 7 40 CC C = 3 2 1 1 10 9 8 120  = OR 3 3 10 3 1 120 C C = ( ) 7 21 7 1E 3 4 5 6 3.924 40 40 120X =  +  +  +  = ( ) 2 2 2 2 2 7 21 7 1E 3 4 5 6 15.724 40 40 120X =  +  +  +  = ( ) ( ) ( ) ( ) 2 22Var E E 15.7 3.9 0.49X X X= − = − = 4. [2017 SRJC MYE/ P2/ Q10 (part of)] (a) It is known that X is a discrete random variable such that ( ) ( ) 24E Var 3 2 1XX = + + . Find E(X). [2] (b) To promote the sales of a snack for children from March to August, one game card is inserted into each box of the snack. For each month, the game cards feature a different series of comic book characters and there are twelve distinct game cards in each series. Andy buys four boxes of the snack in March.

The random variable X denotes the number