6.1
Pr[𝑉 ≥ 1.35] ≤
1
3
6.2
Pr[|T − 30| ≤ 10 ] ≥ 0.84
6.3
a) µk = 8, σ2 = 56
1
b) Pr[𝑘 ≥ 16] =
2
c) Pr[|k − 8| ≥ 8 ] ≤
7
8
d) Pr[(k − 8) ≥ 8 ] ≤
7 15
e) Pr[𝑘 ≥ 16] = � �
8
7
15
6.4
Pr[𝑥 ≥ 2.5] ≤ 0.5
6.5
2
a) Pr �𝑥 ≥ � =
𝜆
2
1
2
b) Pr �𝑥 ≥ � ≤ 1
𝜆
1
1
c) Pr ��𝑥 − � ≥ � ≤
𝜆
2
𝜆
−2
d) Pr �𝑥 ≥ � = 𝑒
𝜆
1
2
6.6
a) Pr[|𝑋| ≥ 2𝜎𝑥 ] ≤
1
2
b) i) Pr[|𝑋| > 2𝜎𝑥 ] = 𝑒 −2 √2
ii) Pr[|𝑋| > 2𝜎𝑥 ] = 0
6.7
Pr[𝑋 − 𝐸[𝑋] ≥ 𝑎] ≤ 0.4235
6.8
|ρ| ≤ 1
6.9
a) Var[Ti ] = 64
b) E[D] = 96, Var[D] = 768
c) Pr[𝐷 > 120] = 0.1848
d) Pr[𝐷 > 120] = 0.1932
e) Pr[𝐷 > 120] ≤ 0.8,
Not a very tight bound
6.10
a) E[Yn ] = 2 , Var[Yn ] =
b) fY (y) =
c) 𝐿 ≥ 30
2
√2π
2
e−2(x−2)
1
4
d) 0.74
6.11
Pr[𝑌 ≥ 0 ] = 𝑄(0) = 0.5
6.12
Pr[Y16 ≤ 50] ≈ 0.57
6.13
17 binders
6.14
Pr[300 < 𝑌800 ≤ 500] ≈ 1 − 2.56 ∗ 10−12 , Pr[400 < 𝑌800 ≤ 440] ≈ 0.49744
6.15
Pr[𝑌1000 ≤ 115] ≈ 0.944
6.16
a) Pr[𝑌 ≤ 3] = 0.0119
b) Pr[𝑌 ≤ 3] ≈ 0.038
6.17
a) Var[θ] =
b) fθ (θ) =
Nπ2
3
1
�2Nπ
3
3
e
−
θ2
2Nπ2
3
6.18
′
a) ∑2n is unbiased
b) ∑2n is the most efficient
6.19
1
a) 𝑝̂ = ∑𝑛𝑘=1 𝑋𝑘
𝑛
b) 𝐸[𝑝̂ ] = 𝑝, 𝑉𝑉𝑉[𝑝̂ ] = 𝑝(1 − 𝑝)
6.20
𝜇̂ 𝑚𝑚 =
𝑛
1
� 𝑋𝑖
𝑛
𝑖=1
6.21
�2 𝑚𝑚 =
𝜎
𝑛
1
� 𝑋𝑖 2
𝑛
𝑖=1
6.22
𝑛
1
𝜇̂ 𝑚𝑚 = � 𝑋𝑖
𝑛
𝑖=1
6.23
P𝑚𝑚 =
𝑘
𝑛
6.24
(2.81, 3.69), (2.73, 3.77)
6.25
6.26