# AMS131, Fall 2014, Section 01: Homeworks

We will refer to the Textbook as DG & S. Problem numbers are taken from the 4th Edition. The homework is a substanstial part of the course, you are expected to try the relative questions after each lecture.

Solutions to the even number questions are in the parentheses.

• Homework 1

Section 1.4: Exercises 6, 7, 8

Section 1.5: Exercises 3, 4 (0.6), 7, 9, 11, 12

Section 1.6: Exercises 1, 3, 4 (1/7), 6 (1/4)

Section 1.7: Exercises  2 (9000), 3, 5, 7, 9

Section 1.8: Exercises 1, 4 (1/10616), 6 (2/n), 7, 9, 11, 14, 17

Section 1.9: Exercises 1, 3, 7, 9

Section 1.10: Exercises 1, 2, (0.85) 7, 10 (1/2)

• Homework 2

Section 2.1: 1, 2, 3, 5, 7, 9 (Answer: 2) P(A|B)=0)

Section 2.2: 1, 2, 4, 5, 7, 10, 13, 15, 23 (Answer: 4) (1/6)^3)

Section 2.3: 4, 5, 7, 8(a), 9 (Answer: 8(a) (0, 0.5870,0.3478,0.0652,0 for coin 1,2,3,4,5))

•  Homework 3

Section 3.1: 2(1/15), 3, 4, 7, Section 5.2: 3, Section 5.3: 1, section 5.4: 3, Section 5.5:1

Section 3.2: 3, 4, 5, 8 , 9, 10, 13 (Answers: 4 a) 0.6458, b) 0.5625, c) 0.5597, 8a) c=2, b) e^(-2)-e(-4), 10 a)c=1/2, b)1-(1/2)^(1/2)

Section 3.3: 3, 4, 5, 6, 8, 9 (Answers: 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, 6) exp(x-3) for x>3, 0 o.w.)

Section 3.4: 2, 3, 5, 7, 9 (Answers 2 a) 0.27 b)0.53 c) 0.69 d)0.3 e) 0.25)

Section 3.5: 1, 2, 3, 6, 7, 8, 10 , 15 (Answers: 2a) 1/15(2x+3), x=0,1,2 1/10(1+y) y=0,1,2,3 b) No; 6) 9/64x^2y^2, b) 0 c)1/2 d) 1/1280; 8) No 10a) 2/pi (-x^2)^(1/2), -1<=x<=1, same for y by symmetry) b) No

Section 3.6: 1, 4, 5, 7, 8 ,9 (Answers: 4 a) f(x|y)=(x+y^2)/(1/2 +y^2) b) 1/3 8) a) 0.264 b) 0.284 c)0.314)

Section 3.7: 1, 2, 3  (Solutions: problem 2: (a) 6, (b) 0.3 if (x2,x3) in {(0,0),(1,1)} and 0.2 if (x2,x3) in {(1,0),(0,1)} (c) 20x1^3(1-x1) if 0< x1< 1 and 0 o.w)

Section 3.8: 1, 3, 4, 5, 7, 8, 13, (Answers: 4) 1/6(4-y)^(1/3), -4<y<4 8) 2y e^(-y^2) )

Section 3.9: 1, 2, 3, 4, 5, 7, 8, 12, 13, 19 (Asnwers: problem 4: g(y)=2(1-y), 0<y<1 problem 8: n>=299)

Section 3.10: 1, 2, 3, 4, 5, 7, 9, 12, 19 (Answers: problem 2: (a) 0.4, (b) 0.49 (c) 0.936; problem 4: (a) 0.4669, (b) 0.4662; problem 12: part (b): it is most likely that C will have the ball at time n+2)

• Homework 4

Section 4.1: 1, 3, 4, 6, 7, 8, 11 (Answers: problem 4: 3.75; problem 6: 2; problem 8: 1/2)

Section 4.2: 2, 3, 7, 8, 9 (Answers: problem 2: -4; problem 8: -8/5)

Section 4.3: 1, 3, 4, 5, 6, 7

Section 4.4: 1, 3, 6, 7, 8, 10 (answers: 6: [exp(tb)-exp(ta)]/t(b-a) prob 8: 3, 2 prob 10: exp(13t^2+t))

Section 4.6: 3, 5, 7, 9, 10, 12, 13 (Answers: problem 12: 245/81)

Section 4.7: 6, 7, 14 (only E(Y|X)) (asnwers: problem 6:0, 14: negative correlated)

Section 5.6: 2, 3, 5, 7, 11, 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 16: 0.8186).

Section 6.2: 5

Section 6.3: 2, 3, 5, 8, 10 (Answers: problem 2: 0.0227; problem 8: 0.0013; problem 10: (a) n>= 1600 (b) n>=106 )