Chap 7
7.20a. P(X  2) = P(2) + P(3) = .4 + .2 = .6
 xP (x) = 0(.1) + 1(.3) + 2(.4) + 3(.2) = 1.7
= V(X) =  ( x  ) P( x ) = (0–1.7) (.1) + (1–1.7) (.3) + (2–1.7)
b.  = E(X) =
2
2
2
7.30  = E(X) =
 2 = V(X) =
2
2
2
(.4) + (3–1.7) (.2) = .81
 xP (x) = 0(.04) + 1(.19) + 2(.22) + 3(.28) + 4(.12) + 5(.09) + 6(.06) = 2.76
 ( x  )
2
2
2
2
P( x ) = (1–2.76) (.04) + (2–2.76) (.19) + (3–2.76) (.28)
2
2
2
+ (4–2.76) (.12) + (5–2.76) (.09) + (6–2.76) (.06) = 2.302
 2  2.302 = 1.517
=
 xP (x) = 1(.24) + 2(.18) + 3(.13) + 4(.10) + 5(.07) + 6(.04) + 7(.04) + 8(.20) = 3.86
= V(X) =  ( x  ) P( x ) = (1–3.86) (.24) + (2–3.86) (.18) + (3–3.86) (.13) + (4–3.86) (.10)
7.32  = E(X) =
2
2
2
2
2
2
2
2
2
2
+ (5–3.86) (.07) +(6–3.86) (.04) + (7–3.86) (.04) + (8–3.86) (.20) = 6.78
 2  6.78 = 2.60
=
7.54 a
b
Refrigerators, x
P(x)
0
.22
1
.49
2
.29
Stoves, y
P(y)
0
.34
1
.39
2
.27
 xP (x) = 0(.22) + 1(.49) + 2(.29) = 1.07
 = V(X) =  ( x  ) P( x ) = (0–1.07) (.22) + (1–1.07) (.49) + (2–1.07) (.29) = .505
d  = E(Y) =  yP( y) = 0(.34) + 1(.39) + 2(.27) = .93
 = V(Y) =  ( y  ) P( y) = (0–.93) (.34) + (1–.93) (.39) + (2–.93) (.27) = .605
e   xyP ( x , y) = (0)(0)(.08) + (0)(1)(.09) + (0)(2)(.05) + (1)(0)(.14) + (1)(1)(.17)
c  x = E(X) =
2
2
2
2
2
y
2
2
2
2
2
all x all y
+ (1)(2)(.18) + (2)(0)(.12) + (2)(1)(0.13) + (2)(2)(.04) = .95
1
COV(X, Y) =
  xyP (x, y) –  x  y = .95 – (1.07)(.93) = –.045
all x all y
 x   2x  .505 = .711,  y   2y  .605 = .778

COV (X, Y )
.045
=
= –.081
xy
(.711)(. 778 )
7.68 a
Stock
1
2
3
Means
.0463
.1293
–.0016
Variances
.0152
.0103
.0040
(Note that if you use the Covariance tool, all entries in the output should
be modified by the factor n/(n-1). The Correlation tool can also be used;
in that case, no correction is necessary.)
b Invest all your money in stock 2; it has the largest mean return.
c Invest all your money in stock 3; it has the smallest variance.
7.70 a
A
Portfolio
of
3
Stocks
1
2
3 Variance-Covariance Matrix
4
5
6
7 Expected Returns
8
9 Weights
10
11 Portfolio Return
12 Expected Value
13 Variance
14 Standard Deviation
B
Stock 1
Stock 2
Stock 3
C
D
E
Stock 1
Stock 2
Stock 3
0.0152
0.0038
0.0103
0.0016
-0.0012
0.0404
0.0463
0.1293
-0.0016
0.5
0.3
0.2
0.0616
0.0062
0.0788
b The mean return on this portfolio is greater than the mean returns on stocks 1 and 3 and the portfolio in
Exercise 7.69, but smaller than that of stock 2. The variance of the returns on this portfolio is smaller than
that for stocks 1 and 2 and larger than that of stock 3 and the portfolio in Exercise 7.69.
7.72 a
2
A
Portfolio of 3 Stocks
1
2
3
4
5
6
7
8
9
10
11
12
13
14
B
Variance-Covariance Matrix
Stock 1
Stock 2
Stock 3
Expected Returns
C
E
Stock 1
Stock 2
Stock 3
0.0973
-0.0293
0.2384
0.0004
0.0247
0.0524
0.0232
0.0601
0.0136
0.3
0.4
0.3
Weights
Portfolio Return
Expected Value
Variance
Standard Deviation
D
0.0351
0.0506
0.2249
b The mean return on the portfolio is greater than the mean returns on stocks 1 and 3, but smaller than that
of stock 2. The variance of the returns on the portfolio is smaller than that for the three stocks.
7.74 a
Stock
1
2
3
4
Means
.0187
–.0176 .0153
.0495
Variances
.0633
.0238
.0532
.0234
b Invest all your money in stock 4; it has the largest mean return.
c Invest all your money in stock 3; it has the smallest variance
7.76 a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
Portfolio of 4 Stocks
Variance-Covariance Matrix
B
Stock 1
Stock 2
Stock 3
Stock 4
Expected Returns
Weights
Portfolio Return
Expected Value
Variance
Standard Deviation
C
D
E
F
Stock 1
Stock 2
Stock 3
Stock 4
0.0633
-0.0012
0.0238
0.0173
-0.0023
0.0234
0.0133
0.0184
-0.0006
0.0532
0.0187
-0.0176
0.0153
0.0495
0.20
0.20
0.10
0.50
0.0265
0.0238
0.1543
b The mean return on this portfolio is greater than the mean return on stocks 1, 2, and 3, and the mean
return on the portfolio in Exercise 7.75. It is smaller than the mean return on stock 4. The variance of the
3
returns on this portfolio is smaller than that for stocks 1, 2, and 4, but larger than the variance on the returns
of stock 3 and the variance of the returns on the portfolio in Exercise 7.75.
7.96 Table 1 with n = 25 and p = .3: P(X
 10) = .902
7.98 Table 1 with n = 25 and p = .75: P(X  15) = 1 – P(X  14) = 1 – .030 = .970
7.104 Excel with n = 100 and p = .20: P(X > 25) = P(X
7.108 Excel with n = 50 and p = .45: P(X
 19) = 1 – P(X  18) = 1 – .12735 = .87265
7.112 a Table 2 with  = 3.5: P(X = 0) = P(X
b Table 2 with  = 3.5: P(X
 26) = 1 – P(X  25) = 1 – .91252 = .08748
 0) = .030
 5) = 1 – P(X  4) = 1 – .725 = .275
c Table 2 with  = 3.5/7: P(X = 1) = P(X
7.116 P(X = 0 with  = 2) =
 1) – P(X  0) = .910 – .607 = .303
e   x e 2 (2) 0
=
= .1353
0!
x!
7.118 P(X = 0 with  = 80/200) =
7.120 a Table 2 with  = 1.5: P(X
e   x e .4 (.4)0
=
=.6703
0!
x!
 2) = 1 – P(X  1) = 1 – .558 = .442
b Table 2  = 6: P(X < 4) = P(X
 3) = .151
7.122 a P(X = 0 with  = 1.5) =
e   x
e 1.5 (1.5) 0
=
.2231
0!
x!
b Table 2 with  = 4.5: P(X
 5) = .703
c Table 2 with  = 3.0: P(X
 3) = 1 – P(X  2 = 1 – .423 = .577
7.126 a P(X = 10 with  = 8) =
e   x
e 8 (8)10
=
= .0993
10!
x!
b Table 2 with  = 8: P(X > 5) = P(X
 6) = 1 – P(X  5) = 1 – .191 = .809
c Table 2 with  = 8: P(X < 12) = P(X
 11) = .888
4
7.130 Table 1 with n = 10 and p = .20: P(X
 6) = 1 – P(X  5) = 1 – .994 = .006
7.132 a Excel with n = 80 and p = .70: P(X > 65) = P(X
 66) = 1 – P(X  65) = 1 – .99207 = .00793
b E(X) = np = 80(.70) = 56
c =
np(1  p)  80(.70)(1  .70) = 4.10
7.136 a  = E(X) =
 2 = V(X) =
 xP (x) = 0(.36) + 1(.22) + 2(.20) + 3(.09) + 4(.08) + 5(.05) = 1.46
 ( x  )
2
2
2
2
P( x ) = (0–1.46) (.36) + (1–1.46) (.22) + (2–1.46) (.20)
2
2
2
+ (3–1.46) (.09) + (4–1.46) (.08) + (5–1.46) (.05) = 2.23
=
 2  2.23 = 1.49
 xP (x) = 0(.15) + 1(.18) + 2(.23) + 3(.26) + 4(.10) + 5(.08) = 2.22
= V(X) =  ( x  ) P( x ) = (0–2.22) (.15) + (1–2.22) (.18) + (2–2.22) (.23)
b  = E(X) =
2
2
2
2
2
2
2
2
+ (3–2.22) (.26) + (4–2.22) (.10) + (5–2.22) (.08) = 2.11
=
 2  2.11 = 1.45
c Viewers of nonviolent shows remember more about the product that was advertised.
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