Exercise 7.1 | Q 14 | Page 233 Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the standard deviation of X. SOLUTION If two fair dice are rolled then the sample space S of this experiment is S= {(1,1), (1,2),(1,3).(1,4),(1,5),(1,5), (1,6),.(2,1),(2,2),(2,3). (2,3),(2,4),(2,5),(2,6),(3,1), (3,2),(3,3), (3, 4),(3,5), (3,6), (4,1),(4,2), (4,3), (4,4),(4,5), (4,6),(5,1), (5,2),(5,3), (5,4),(5,5), (5,6), (6,1), (6,2),( 6,3).(6,4),(6,5), (6,6)} : n(S) = 36 Let X denote the sum of the numbers on uppermost faces. Then X can take the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 sum of Nos. (x) 2 3 4 5 6 7 8 9 Favourable events (1,1) (1, 2), (2, 1) (1, 3), (2, 2), (3, 1) (1, 4), (2, 3), (3, 2), (4, 1) (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) (3, 6), (4, 5), (5, 4), (6, 3) No of favourable P (x) 1 2 3 4 5 6 5 1 36 2 36 3 36 4 36 5 36 6 36 5 36 4 36 2 P[X=x1] : the probability distribution of X is given by X=x; 2 3 10 11 36 X 12 1 36 ()+=() 36 Expected value = E (X) = Exi- P (x) +3 1 1 4 X +9x 36 36 36 36 Also, Ex- P (x) x 252 =7. +144 x 2 36 1 4 5 3 (4, 6), (5, 5), (6, 4) + 16 × 36 (5, 6), (6, 5) 4 36 36 (6,6) + 25 x 6 7 5 36 =(2+6+ 12 + 20 + 30 + 42 + 40 + 36 + 30 + 22 + 12) 4 6 36 8 5 36 36 5 +3 36 × 36 4 10 36 36 3 6 + 49 x 36 11 2 +64 x 3 +)+0)+)+) +)+)+ 1o)+11 2 1 5 12 36 36 1 4 + 81 x 36 3 36 36 2 36 1 36 + 100 3 36 +121 1 36 1 36 4 + 18 + 48 + 100 + 180 + 294 + 320 + 324 + 300 + 242 + 144] :. variance = V) =Ex. P (X) - [E(C)] (1974) = 54.83 = 54-83 - 49 = 5.83 :. standard deviation = = V5.83 = 2.41 Exercise 7.1 |Q15| Page 233 X A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X. SOLUTIOM There are 15 students in the class. Each student has the same chance to be chosen. Therefore, the probability of each student to be selected is 1/15 The given information can be compiled in the frequency table as follows. 14 2 15 1 16 2 17 3 18 1 19 2 20 3 21 1 P(X = 14) =P(X = 15) =PX= 16) = P(X = 16) =P(X = 18) = P(X = 19) =PX= 20) =P(X = 21) = EX, POX) Therefore, the probability distribution of random variable X is as follows. X 263 Then, mean of X= E(X) 15 = = 17.53 1 3 2 15 14 EX) - x? POX) 2 15 4683 15 = 312.2 (28 + 15 + 32 + 51+ 18 + 38+ 60 + 21) 15 15 1 = 312.2 - (17.53)2 15 - (14) x +(15) x + (16)x +(17) x- 15 = 312.2 -307.3 = 4.9 Standard deviation = 16 2 =14 x + 15 x t 16 x+17 x +18 x +19 x+20 xE +21x E 15 15 3 2 Variance = VØ) =Ex. P(x) - (E(X1² 15 2 Exercise 7.1 |Q 16 | Page 233 3 17 15 3 15 (392 + 225 + 512 + 867 + 324 + 722 + 1200 + 441) 2 /vx)- Va9 - 221 + (18)² x 18 1 15 15 3 19 Hence, mean = 17.53, variance = 4.9 and standard deviation = 2.21. 2 15 + (19)x + (20) x - 3 15 20 3 15 + (21)°x In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X=0 if he opposed, and X=1 if he is in favour. Find E(X) and Var(X). 1 15 2 1 15 SOLUTION It is given that P(X = 0) = 30% = 30/100 = 0.3 P(X = 1) = 70% = Therefore, the probability distribution is as follows. Then, E(0) = XP(x) =0 x 0.3+1 x 0.7 = 0.7 E(X2) = X,P(x) = 0.7 = 0²x 0.3 + (1) x 0.7 X P(X) = 0.7- (O.7)2 = 0.7 - 0.49 = 0.21 70 100 It is known that, var (0) = E (X3) - [E(X)]2 =0.7 Hence, E(X) = 0.7 and Var(X) = 0.21. 0 0.3 1 0.7