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7. Notes

Books

This chapter contains core topics that are discussed in most books on probability.

• An Introduction to Probability Theory and It's Applications, Volume 1 (3rd edition) by William Feller is considered one of the best references on probability, and indeed one of the best probability books ever written.
• An excellent book on elementary probability with a wealth of examples and exercises is A First Course in Probability (5th edition) by Sheldon Ross
• A compact treatment of elementary probability is The Essentials of Probability by Richard Durrett
• For a higher level measure-theoretic treatment of probability, see Probability and Measure, by Patrick Billingsley.
• A good treatment of the early history of probability is Games, Gods and Gambling, by Florence David

External sites

• The definitive site for the historical information about probability is the History of Mathematics site.

Selected Answers for Section 1

1.4. Let X denote the score. E(X) = 7/2.

1.6. Let X denote the score. E(X) = 7/2.

1.7. E(X) = 3/5.

1.21. Let Y = X².

1. g(y) = (1/4)y ^-1/2 for 0 < y < 1, g(y) = (1/8)y ^-1/2 for 1 < y < 9.
2. E(Y) = 7/3.
3. E(Y) = 7/3.

1.22. Let Y = X².

1. E(X) = 18 / 5

2. y	1	4	9	16	25
   P(Y = y)	1/30	2/15	3/20	4/15	5/12

3. E(Y) = 83 / 5
4. E(Y) = 83 / 5

1.23.

1. E(1/X) = 2
2. E(X^1/2) = 48 / 63

1.24.

1. E(X) = 5 / 12
2. E(Y) = 3 / 4
3. E(X²Y) = 7 / 36
4. E(X² + Y²) = 5 / 6.

1.32.

1. E(Y) = 7
2. E(Z) = 49 / 4
3. E(U) = 101 / 36
4. E(V) = 19 / 4

1.33. E(3X + 4Y - 7) = 0

1.34. E[(3X - 4)(2Y + 7)] = 33

1.35. Let N denote the number of ducks killed.

E(N) = 10[1 - (9/10)⁵] = 4.095

1.36. E(Xⁿ) = (b^{n + 1} - a^{n + 1}) / [(n + 1)(b - a)]

1.37. E(Xⁿ) = 12[1 / (n + 3) - 1 / (n + 4)]

1.44.

1. E(X) = 1 / r
3. exp(-rt) < 1 / rt for t > 0

1.45.

1. E(Y) = 1 / p
3. (1 - p)^{n - 1} < 1 / np for n = 1, 2, ...

1.50.

1. E(X) = a / (a - 1)
2. E(1/X) = a / (a + 1)
4. a / (a + 1) > (a - 1) / a

1.53.

1. E(X² + Y²) = 5 / 6
2. [E(X)]² + [E(Y)]² = 53 / 72

1.54. E(X | X > t) = t + 1 / r.

1.56. E(Y | Y is even) = 2(1 - p)² / [p(2 - p)³]

1.57. E(XY | Y > X) = 1/3.

Selected Answers for Section 2

2.9. Let X denote the die score.

1. E(X) = 7/2
2. var(X) = 35/12
3. sd(X) ~ 1.708

2.11. Let X denote the die score.

1. E(X) = 7/2
2. var(X) = 15/4
3. sd(X) ~ 1.936

2.22.

1. var(3X - 2) = 36
2. E(X²) = 29

2.24. z = 8.53.

2.27. E(Y) = 4/3, sd(Y) = 2/3, k = 2

1. P[|Y - E(Y)| k sd(Y)] = 1/16.
2. 1 / k² = 1/4

2.28. E(X) = 1 / r, sd(Y) = 1 / r.

1. P[|X - E(X)| k sd(Y)] = exp[-(k + 1)]
2. 1 / k².

2.32.

1. E(X) = 1/2, var(X) = 1/20, skew(X) = 0, kurt(X) = 15/7
2. E(X) = 3/5, var(X) = 1/25, skew(X) = -2/7, kurt(X) = 33/14
3. E(X) = 2/5, var(X) = 1/25, skew(X) = 02/7, kurt(X) = 33/14

2.38.

1. ||X||_k = 1 / (k + 1)^1/k.
3. 1

2.39.

1. ||X||_k = [a / (a - k)]^1/k if k < a, ||X||_k = if k a.
3. 

2.40.

1. ||X + Y||_k = [(2^k+3 - 2) / (k + 3)(k + 2)]^1/k.
2. ||X||_k + ||Y||_k = 2[1 / (k + 2) + 1 / 2(k + 1)]^1/k.

2.48.

1. When p < 1/2, the minimum of E[|I - t|] is p and occurs at t = 0.
2. When p = 1/2, the minimum of E[|I - t|] is 1/2 and occurs for t in [0, 1].
3. When p > 1/2, the minimum of E[|I - t|] is 1 - p and occurs at t = 1.

Selected Answers for Section 3

3.14.

1. cov(X₁, X₂) = 0, cor(X₁, X₂) = 0
2. cov(X₁, Y) = 35 / 12, cor(X₁, Y) = 2^-1/2 ~ 0.7071.
3. cov(X₁, U) = 35 / 24, cor(X1, U) ~ 0.6082
4. cov(U, V) = 1369 / 1296, cor(U, V) = 1369 / 2555 ~ 0.5358
5. cov(U, Y) = 35 / 12, cor(U, Y) = 0.8601

3.15. cov(2X - 5, 4Y + 2) = 24.

3.16.

1. cov(X, Y) = -1 / 144.
2. cor(X, Y) = -1 / 11 ~ 0.0909

3.17.

1. cov(X, Y) = 1 / 48.
2. cor(X, Y) ~ 0.4402

3.18.

1. cov(X, Y) = 0.
2. cor(X, Y) = 0.

3.19.

1. cov(X, Y) = 5 / 336
2. cor(X, Y) ~ 0.0.5423

3.24. var(2X + 3Y - 7) = 83

3.25. var(3X - 4Y + 5) = 182

3.27. Let Y denote the sum of the dice scores.

1. E(Y) = 7n / 2.
2. var(Y) = 35n / 12.

3.32.

1. cov(A, B) = 1 / 24.
2. cor(A, B) ~ 0.1768.

3.33.

1. Y* = (7 - X) / 11
2. X* = (7 - Y) / 11
3. cor²(X, Y) = 1 / 121 = 0.0083

3.40.

1. Y* = (26 + 15X) / 43
2. X* = 5Y / 9
3. cor²(X, Y) = 25 / 129 ~ 0.1938

3.41.

1. Y* = 2 / 3
2. X* = 3 / 4
3. cor²(X, Y) = 0

3.42.

1. Y* = (30 + 20X) / 51
2. X* = 3Y / 4
3. cor²(X, Y) = 5 / 17 ~ 0.2941

3.43.

1. Y* = 7 / 2 + X₁.
2. U* = 7 / 9 + X₁ / 2.
3. V* = 49 / 19 + X₁ / 2.

3.53. <X, Y> = 1/3

1. ||X||₂ ||Y||₂ = 5 / 12.
2. ||X||₃ ||Y||_3/2 ~ 0.4248.

Selected Answers for Section 4

4.32.

1. M(s, t) = 2[exp(s + t) - 1] / [s(s + t)] - 2[exp(t) - 1] / (st) for s, t 0
2. M_X(s) = 2[exp(s) / s² - 1 / s² - 1 / s] for s 0.
3. M_Y(t) = 2[t exp(t) - exp(t) + 1] / t² for t 0.
4. M_{X + Y}(t) = [exp(2t) - 1] / t² - 2[exp(t) - 1] / t² for t 0.

4.33.

1. M(s, t) = {exp(s + t)[-2st + s + t] + exp(t)[st - s - t] + exp(s)[st - s - t] + s + t} / (s² t²) for s, t 0.
2. M_X(s) = [3s exp(s) - 2 exp(s) - s + 2] / (2s²) for s 0.
3. M_Y(t) = [3t exp(t) - 2 exp(t) - t + 2] / (2t²) for t 0.
4. M_{X + Y}(t) = 2[exp(2t) (-t + 1) + exp(t)(t - 2) + 1] / t³ for t 0.

Selected Answers for Section 5

5.13. E(Y | X) = 0.

5.15. E(Y | X) = (X + 6) / 2.

5.17.

1. E(Y | X) = (3X + 2) / (6X + 3)
2. E(X | Y) = (3Y + 2) / (6Y + 3)

5.18.

1. E(Y | X) = (5X² + 5X + 2) / (9X + 3)
2. E(X | Y) = 5Y / 9

5.19.

1. E(Y | X) = 2 / 3.
2. E(X | Y) = 3 / 4.

5.20.

1. E(Y | X) = 2(X² + X + 1) / 3(X + 1)
2. E(X | Y) = 3Y / 4.

5.21.

1. E(Y | X₁) = 7 / 2 + X₁.

2. x	1	2	3	4	5	6
   E(U | X1 = x)	1	11/6	5/2	3	10/3	7/2

3. u	1	2	3	4	5	6
   E(Y | U = u)	52/11	56/9	54/7	46/5	32/3	12

4. E(X₂ | X₁) = 7/2

5.22. E[exp(X) Y - sin(X) Z | X] = X³ exp(X) - sin(X) / (1 + X²)

5.24. P(H) = 1/2

5.28.

1. Y* = (7 - X) / 11.
2. E(Y | X) = (3X + 2) / (6X + 3)

5.29.

1. Y* = (26 + 15X) / 43
2. E(Y | X) = (5X² + 5X + 2) / (9X + 3)

5.30.

1. Y* = 2 / 3
2. E(Y | X) = 2 / 3.

5.31.

1. Y* = (30 + 20X) / 51
2. E(Y | X) = 2(X² + X + 1) / 3(X + 1)

5.34.

1. var(Y) = 11 / 144 ~ 0.0764.
2. var(Y)[1 - cor²(X, Y)] = 5 / 66 ~ 0.0758.
3. var(Y) - var[E(Y | X)] = 1 / 12 - ln(3) / 144 ~ 0.0757

5.35.

1. var(Y) = 3 / 80 ~ 0.0375
2. var(Y)[1 - cor²(X, Y)] = 13 / 430 ~ 0.0302
3. var(Y) - var[E(Y | X)] = 1837 / 21870 - 512 ln(2) / 6561 ~ 0.0299

5.36.

1. var(Y) = 1 / 18
2. var(Y)[1 - cor²(X, Y)] = 1 / 18
3. var(Y) - var[E(Y | X)] = 1 / 18

5.37.

1. var(Y) = 5 / 252 ~ 0.0198
2. var(Y)[1 - cor²(X, Y)] = 5 / 357 ~ 0.0140
3. var(Y) - var[E(Y | X)] = 292 / 63 - 20 ln(2) / 3 ~ 0.0139

5.38.

1. E(Y | X) = X / 2.
2. var(Y | X) = X² / 12.
3. var(Y) = 7 / 144.

5.44.

1. Given N, X has the binomial distribution with parameters N and p = 1/2.
2. E(X | N) = N / 2.
3. var(X | N) = N / 4.
4. E(X) = 7 / 4
5. var(X) = 7 / 3.

5.46. Let Y denote the amount of money spent during the hour.

1. E(Y) = $1000
2. sd(Y) ~ $30.822

5.51. Let X denote the die score

1. E(X) = 7 / 2.
2. var(X) = 1.8634

Selected Answers for Section 6

6.17.

1. E(X, Y)	7 / 12
   	7 / 12

2. VC(X, Y)	11 / 144	-1 / 144
   	-1 / 144	11 / 144

6.18.

1. E(X, Y)	5 / 12
   	3 / 4

2. VC(X, Y)	43 / 720	1 / 48
   	1 / 48	3 / 80

6.19.

1. E(X, Y)	3 / 4
   	2 / 3

2. VC(X, Y) =	3 / 80	0
   	0	1 / 18

6.20.

1. E(X, Y)	5 / 8
   	5 / 6

2. VC(X, Y)	17 / 448	5 / 336
   	5 / 336	5 / 252

6.21.

1. E(X, Y, Z)	1 / 4
   	1 / 2
   	3 / 4

2. VC(X, Y, Z)	3 / 80	1 / 40	1 / 80
   	1 / 40	1 / 20	1 / 40
   	1 / 80	1 / 40	3 / 80

6.22.

1. E(X, Y)	1 / 2
   	1 / 4

2. VC(X, Y)	1 / 12	1 / 24
   	1 / 24	7 / 144

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