MATH 3670N Quiz 2
Name:
Key
Problem 1
Suppose a random variable
X
has pdf
f
(
x
) =
2
x
3
, x
≥
1
0
,
else.
a.
(3 points) Find
E
(
X
)
.
E
(
X
) =
R
∞
1
2
x
2
dx
= lim
b
→∞
R
b
1
2
x
2
dx
= lim
b
→∞
(

2
x
)
b
1
= 2
.
b.
Bonus (1 point) Does the variance of
X
exist? Explain.
E
(
X
2
) =
R
∞
1
2
x
dx
= lim
b
→∞
2 ln(
b
) =
∞
.
Hence
σ
2
=
∞ 
4 =
∞
.
So the
variance is infinite and hence does not exist as a real number.
Problem 2
(4 points) Suppose
X
is an exponential random variable of mean
μ
= 2
,
that is
X
has pdf
f
(
x
) =
1
2
e

x/
2
, x >
0
0
,
else.
Find
P
(

X

μ

<
2)
.
P
(

X

2

<
2) =
P
(0
< X <
4) =
R
4
0
1
2
e

x/
2
dx
= 1

1
e
2
.
Problem 3
(4 points) Suppose
X
is normally distributed with mean 3 and stan
dard deviation 2
.
Find
P
(

1
≤
X <
1)
.
You may use Table 1 on the next page.
P
(

1
≤
X <
1) =
P
(

2
<
X

3
2
<

1) =
P
(

2
< Z <

1) =
φ
(

1)

φ
(

2) =
(1

φ
(1))

(1

φ
(2)) =
φ
(2)

φ
(1)
≈
0
.
97725

0
.
84134 = 0
.
13591
.
STANDARD NORMAL DISTRIBUTION: Table Values Represent AREA to the LEFT of the Z score.
Z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
.50000
.50399
.50798
.51197
.51595
.51994
.52392
.52790
.53188
.53586
0.1
.53983
.54380
.54776
.55172
.55567
.55962
.56356
.56749
.57142
.57535
0.2
.57926
.58317
.58706
.59095
.59483
.59871
.60257
.60642
.61026
.61409
0.3
.61791
.62172
.62552
.62930
.63307
.63683
.64058
.64431
.64803
.65173
0.4
.65542
.65910
.66276
.66640
.67003
.67364
.67724
.68082
.68439
.68793
0.5
.69146
.69497
.69847
.70194
.70540
.70884
.71226
.71566
.71904
.72240
0.6
.72575
.72907
.73237
.73565
.73891
.74215
.74537
.74857
.75175
.75490
0.7
.75804
.76115
.76424
.76730
.77035
.77337
.77637
.77935
.78230
.78524
0.8
.78814
.79103
.79389
.79673
.79955
.80234
.80511
.80785
.81057
.81327
0.9
.81594
.81859
.82121
.82381
.82639
.82894
.83147
.83398
.83646
.83891
1.0
.84134
.84375
.84614
.84849
.85083
.85314
.85543
.85769
.85993
.86214
1.1
.86433
.86650
.86864
.87076
.87286
.87493
.87698
.87900
.88100
.88298
1.2
.88493
.88686
.88877
.89065
.89251
.89435
.89617
.89796
.89973
.90147
1.3
.90320
.90490
.90658
.90824
.90988
.91149
.91309
.91466
.91621
.91774
1.4
.91924
.92073
.92220
.92364
.92507
.92647
.92785
.92922
.93056
.93189
1.5
.93319
.93448
.93574
.93699
.93822
.93943
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 Fall '14
 KimM.Cobb
 Statistics, Normal Distribution, Variance, standard normal distribution