Introduction to Probability Theory

Howworks will not be collected or graded. Homeworks will give you a very close indication of the material that will be covered in exams and quizzes. Some homewok will be reviewed in the sections (Note: Answers to even problems are in parentheses).

**Homework 1**

Section 1.4: Exercises 1, 3, 6, 7, 8

Section 1.5: Exercises 3, 4 (0.6), 6 (0.5), 7, 9, 11, 14 ((a) Anti A 0.38 Anti B 0.16 (b) 0.04)

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

Section 1.7: Exercises 2 (9000), 3, 5, 7, 8 (360/2401)

Section 1.8: Exercises 1, 4 (1/10626), 6 (2/n), 10 (0.1630), 11, 12 (0.5798), 17

**Homework 2**

Section 2.1: 1, 2, 3, 4, 5, 6, 9 (problem 2: 0; problem 4: 1/27).

Section 2.2: 1, 2, 4, 7, 10, 11, 12, 13, 23 (problem 4: 0.00463; problem 12: (a) 1/4 (b) 11/24).

Section 2.3: 3, 4, 5, 6, 7 (problem 4: 0.00019, problem 6: a) 96/97, b) 64/65)

**Homework 3**

Section 3.1: 1, 2, 5, 6, 7, 8, 10 (problem 1: 6/11; problem 2: 1/15, problem 6: 0.1509, problem 8: 0.1330).

Section 5.2: 8 (0.6993)

Section 5.3: 2 (Solutions: Problem 2: 0.4266).

Section 5.4: 3.

Section 5.5: 1.

Section 3.2: 3, 4, 5, 7, 8, 10 (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 (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).

Section 3.4: 1, 2, 3, 4, 5, 8, 10 (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)

Section 3.5: 2, 3, 4, 6, 7, 8, 11 (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.

** Homework 4**

Section 3.6: 1, 4, 5, 6, 8 (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)).

Section 3.9: 1, 2, 6, 7, 8

Section 3.10: 2, 3, 4, 5, 6, 7, 8, 9, 19 (problem 2: (a) 0.4 (b) 0.49 (c) 0.936; problem 4: (a) 0.4669 (b) 0.4662; problem 6: (a) 1/(n^p), (b) Pr(X even)=1/(n^2), Pr(X odd)=1-1/(n^2), problem 8: (a) 0.29, (b) 0.6239)

**Homework 5**

Section 3.11: 8, 14, 16, 17 (problem 8: F(x)[1+F^2(x)-F^3(x)], 14:2/3, problem 16: (a) 5/8, (b) 2(x+y)/(1+2x-3x^2))

Section 4.1: 3, 6, 8 (answer: 1/2), 11

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: 3.

Section 5.4: 6, 7 (Solutions: Problem 6: exp(-3.6))

Section 5.5: 1, 2, 3 (Solutions: Problem 2: (a) 145, (b) 4350).

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 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).