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4.3 The Normal Distribution
247
% b
W1 + · · · + Wn − nμ
1
2
e−z /2 dz
lim P a ≤
≤b = √
√
n→∞
nσ
2π a
Proof See Appendix 4.A.2.
Comment The central limit theorem is often stated in terms of the average of W1 ,
W2 , . . ., and Wn , rather than their sum. Since
1
(W1 + · · · + Wn ) = E(W ) = μ and
E
n
1
Var (W1 + · · · + Wn ) = σ 2 /n,
n
Theorem 4.3.2 can be stated in the equivalent form
% b
W −μ
1
2
e−z /2 dz
lim P a ≤
√ ≤b = √
n→∞
σ/ n
2π a
We will use both formulations, the choice depending on which is more convenient
for the problem at hand.
Example
4.3.2
The top of Table 4.3.2 shows a Minitab simulation where forty random samples of
size 5 were drawn from a uniform pdf defined over the interval [0, 1]. Each row
corresponds to a different sample. The sum of the five numbers appearing in a given
sample is denoted “y” and is listed in column C6. For this particular uniform pdf,
1
(recall Question 3.6.4), so
μ = 12 and σ 2 = 12
W1 + · · · + Wn − nμ Y − 52
= /
√
nσ
5
12
Table 4.3.2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
C1
y1
C2
y2
C3
y3
C4
y4
C5
y5
C6
y
C7
Z ratio
0.556099
0.497846
0.284027
0.599286
0.280689
0.462741
0.556940
0.102855
0.642859
0.017770
0.331291
0.355047
0.626197
0.211714
0.535199
0.810374
0.646873
0.588979
0.209458
0.667891
0.692159
0.349264
0.246789
0.679119
0.004636
0.568188
0.410705
0.961126
0.304754
0.404505
0.130715
0.153955
0.354373
0.272095
0.414743
0.194460
0.036593
0.471254
0.719907
0.559210
0.728131
0.416351
0.118571
0.920597
0.530345
0.045544
0.603642
0.082226
0.673821
0.956614
0.614309
0.839481
0.728826
0.613070
0.711414
0.014393
0.299165
0.908079
0.979254
0.575467
0.933018
0.213012
0.333023
0.827269
0.233126
0.819901
0.439456
0.694474
0.314434
0.489125
0.918221
0.518450
0.801093
0.075108
0.242582
0.585492
0.675899
0.520614
0.405782
0.897901
2.46429
3.13544
1.96199
2.99559
2.05270
2.38545
3.15327
1.87403
2.47588
1.98550
2.08240
3.39773
3.07021
1.39539
2.00836
2.77172
−0.05532
0.98441
−0.83348
0.76777
−0.69295
−0.17745
1.01204
−0.96975
−0.03736
−0.79707
−0.64694
1.39076
0.88337
−1.71125
−0.76164
0.42095