# AMS131, Fall 2011, Section 01: Homework Assignments

HOMEWORK ASSIGNMENT #1

• AMS 131 problems. Section 1.4: 1,2,3,6,7,8; Section 1.5: 3,4,5,6,7,8,9,11,14 (Solutions: problem 4: 0.6; problem5: 0.4, problem 6: 0.5; problem 8: 0.3; problem 14: (a) 0.16, (b) 0.04); Section 1.6: 1,2,3,4,5,6 (solutions: problem 2: 0.5; problem 3: 2/3; problem 4: 1/7; problem 6: 1/4); Section 1.7: 2,3,4,5,6,7,8,9 (Solutions: problem 2: 9000; problem 3: 120; problem 4: 24; problem 6: 0.01543; problem 8: 360/2401; problem 9: 1/20) .
• Additional 203 problems. Section 1.4: 4,5; Section 1.5: 12,13; Section 1.7: 10 (Solution for (a), (b) and (c): r/100).

HOMEWORK ASSIGNMENT #2

• AMS 131 problems. Section 1.8: 4,6,7,9,10,11,12,13,17,18 (Solutions: problem 4: 1/10626; problem 6: 2/n; problem 10: 0.1630; problem 12: 0.5798; problem 17: C13,4/C52,4; problem 18: approx 0.0143); Section 1.9: 1,2,3,4,7,8 (Solutions: problem 2: 50!/(18! 12! 8! 12!); problem 4: 1/50400; problem 8: 0.0324);  Section 1.10: 2, 3 (Solutions: problem 2: 0.85; problem 3: 45).
• Additional 203 problems. Section1.8: 14,15,18 (Solution: approx 0.0143),20. Section 1.10: 7,13.

HOMEWORK ASSIGNMENT #3

• Section 2.1: 1, 2, 3, 4, 5, 6, 9 (Solutions: problem 2: 0; problem 4: 1/27). Section 2.2: 1, 2, 4, 7, 10, 11, 12, 13, 23 (Solutions: problem 4: 0.00463; problem 12: (a) 1/4 (b) 11/24). Section 2.3 (old edition of the book): 2, 3, 4, 7, 8, 9, 10 (Solutions: problem 2: 0.056; problem 3: 0.47; problem 4: 5/12; problem 8: 0.00019; problem 10: (a) 96/97 (b) 64/65). Section 2.3 (new edition of the book): 3,4,5,6,7 (Solutions: problem 4: 0.00019, problem 6: a) 96/97, b) 64/65)
• Additional 203 problems. Section 2.1: 11,12,13. Section 2.2: 14( (Solution: 1-0.99^10), 15 (161 or more), 19, 23. Section 2.3: 8, 16 (solution: 8(a) 0, 0.587,0.3478,0.0652,0; 8(b) 0.1291; 16 (b) 0.119).

HOMEWORK ASSIGNMENT #4

• Section 3.1: 1,2,5,6,7,8,10 (Solutions: problem 1: 6/11; problem 2: 1/15, problem 6: 0.1509, problem 8: 0.1330). Section 3.2: 3,4,5,7,8,10 (Solutions: problem 4: (a) 3/7; (b)37/56; problem 8: (a) 2 (b) exp(-2)-exp(-4); problem 10: (a) 1/2; (b) 1-(1/2)^(1/2). Section 3.3: 2, 3, 4, 5, 6, 9 (Solutions: problem 4: (a) 0.1, (b) 0.1, (c) 0.2, (d) 0, (e) 0.6, (f) 0.4, (g) 0.7, (h) 0, (i) 0, (j) 0, (k) 0, (l) 0.2).
• Additional 203 problems. Section 3.1: 11. Section 3.2: 9,10 (sol: (a) 1/2, (b) 0.293). Section 3.3: 8,10,17(a).

HOMEWORK ASSIGNMENT #5

• Section 3.4: 1,2,3,4,5,8,10 (Solutions: problem 1: (a) 1/2 (b) 0.75; problem 2: (a) 0.27, (b) 0.53, (c) 0.69, (d) 0.3, (e) 0.25; problem 3: (a) 1/40 (b) 1/20 (c) 7/40 (d) 0.7; problem 4 (a) 3/2 (b) 3/8 (c) 1/8 (d) 1/2 (e) 0; problem 5: (a) 5/4; (b) 79/256 (c) 13/16 (d) 0; problem 8: (a) 5/78; (b) 25/78; (c) y(9+y)/52 (d) (3x^2+2y)/156); problem 10: (b) 1/3)

HOMEWORK ASSIGNMENT #6

• Section 3.5: 2, 3, 4, 6, 7, 8, 11 (Solutions: problem 2: (a) (2x+3)/15 (b) (1+y)/10; problem 3: (b) they are independent; problem 4 (a) f(x)=15 x^2(1-x^2)/4 and f(y) = (5(1-y)^{3/2})/2 (b) not independent; problem 7: yes, problem 6: (b) 0 (c) 1/2; (d) 1/1280; problem 8: no; problem 11: 11/36.  Section 3.6: 1, 4, 5, 6, 8 (Solutions: problem 4: (b) 1/3; problem 6: (b) 1/3; problem 8: (a) 0.264, (b) 0.284, (c) 0.314) .  Section 3.8: 1, 2, 3, 8, 9, 13 (Solutions: problem 8: 2y exp( -y^2)).
• Additional 203 problems: Section 3.5: 10, 13, 15 (sol: problem 10: (a) 2(1-x)^{1/2}/pi for -1 <= x <= 1 and 0 otherwise; (b) not independent. Section 3.6: 3, 12 (sol: problem 12: (a) 2^x/3^{x+1}, x=0,1,... (b) 3 exp(-3y) for y >0 (c) 9y exp(-3y) for y>0. (d) the ratio is greater if y> 1/3 . Section 3.7: 1, 2, 3, 8 (sol: problem 2 (a) c=6 (b) 0.3 if (x2 , x3 ) in {(0, 0), (1, 1)} and 0.2 if (x2 , x3 ) in {(1, 0), (0, 1)}, (c) 20 x1^3 (1-x1) if 0 < x1< 1 and 0 otherwise; problem 8: (a) exp(-m)/n! with m=max{y1,...,yn} and (b) exp(-(x-m)) for x>m and 0 otherwise).

HOMEWORK ASSIGNMENT #7

• Section 3.9: 1, 2, 6, 7, 8 (Solutions: problem 8: n>=299). Section 3.10 new edition: 1,2,3,4,5 (Solutions: problem 2: (a) 0.4 (b) 0.49 (c) 0.936; problem 4: (a) 0.4669 (b) 0.4662). Note: These problems correspond to problems 1,2,3,4, and 5 in Section 2.4 of the old edition. Section 3.11 new edition: 8, 14, 16, 17 (Solutions: problem 8: F(x)[1+F^2(x)-F^3(x)] 14: 2/3).Note: These problems correspond to problems 8,14,16,and 17 in Section 3.10 of the old edition. Section 4.1: 3, 6, 7,8, 11 (Solutions: problem 6: 2; problem 8: 1/2).
• Additional 203 problems: Section 3.9: 4,8 (Solutions: problem 4: the  pdf of Y is 2(1-y) for 0 < y < 1 and 0 otherwise; problem 8: n>=299). Section 3.10 new edition: 12 (Solution: part (b): it is most likely that C will have the ball at time n+2). Note: this problem corresponds to problem 12 in Section 2.4 of the old edition. Section 3.11 new edition : 23. Note: this problem corresponds to problem 23 in Section 3.10 of the old edition.

HOMEWORK ASSIGNMENT #8

• Section 4.2:  2, 3, 7, 9 (Solutions: problem 2: -4); Section 4.3: 1, 3, 4, 6, 7. Section 4.6: 5, 9, 12, 13 (Solutions: problem 12: 245/81; problem 13: (a) 11, (b) 51). Section 4.7: 2, 6, 7 (Solutions: problem 2: 80.8, problem 6: 0). Section 5.3: 2, 3 (Solutions: Problem 2: 0.4266). Section 5.4: 6, 7 (Solutions: Problem 6: exp(-3.6)). Section 5.5: 2, 3 (Solutions: Problem 2: (a) 145, (b) 4350).
• Additional 203 problems. Section 4.2: 5. Section 4.3: 9. Section 4.4: 7,9.

HOMEWORK ASSIGNMENT #9

• Section 4.6: 5, 9, 12, 13 (Solutions: problem 12: 245/81; problem 13: (a) 11, (b) 51). Section 4.7: 2, 6, 7 (Solutions: problem 2: 80.8, problem 6: 0). Section 5.2: 8, 10 (Solutions:  Problem 8: 0.6993, Problem 10: np/(1-q^n)). Section 5.6: 1, 2, 3, 4, 5, 11, 15, 16 (Solutions: problem 2 (a) 0.8413 (b) 0.4013 (c) 0 (d) 0.2858 (e) 0.6915 (f) 0.2426 (g) 0.6247 (h) 0.4599; problem  3: 400/81 problem 4: the 25th quantile is 65.30 and the 75th is 70.698; problem 5: 0.9960; problem 11: n=1083; problem 16: 0.8186). Section 5.7: 8,10. Section 6.2: 5. Section 6.3:  2,3,5,8,9,10 (Solution:  problem 2: 0.0227; problem 3:  0.9938; problem  5:  542;  problem 8: 0.0013, problem 9: (a) 0.36, (b) 0.7888, problem 10: (a) n>=1600 (b) n>= 106).