Section 8.3
You will need the equations from section 8.2 with the following…
For this section we add two equations (that are very similar to one another):
First, the definition formula for the variance…the equation is:
2 = ∑((X-)2 * P(X) )
Which means that you subtract the mean from each value of X, and then
square that, and then that multiple each of the X’s in the probability
distribution by its probability, and then add them up…giving you the variance.
Second, the computing formula for the variance…the equation is:
2 = E(X2) - 2
Which means that you take the mean and square it, and then subtract that from
the expected value of X-squared…giving you the variance.
The standard deviation (which is ultimately what you really want to know is
simply the square root of the variance (no matter how you get it).
If you do not like equations, each will be done in a table as well…
Problem #55
µ = 2*0.6 + 3*0.4 = 2.4
variance by definition formula:
2 = (2-2.4)2*0.6 + (3-2.4)2*0.4 = 0.24
(standard deviation) 
= sqrt(0.24) = 0.4899
variance by computing formula:
E(x2) = 22*0.6 + 32*0.4 = 6
2 = 6 - (2.4)2 = 0.24
(standard deviation) 
OR
= sqrt(0.24) = 0.4899
variance by computing formula:
x*P(x)
x
2*0.6 = 1.2
2
3*0.4 = 1.2
3
1.2+1.2 = 2.4
P(x)
0.6
0.4
v2
4
9
v2*P(v)
4*0.6 = 2.4
9*0.4 = 3.6
2.4+3.6 = 6
(v-µ)2
0.16
0.36
(v-µ)2*P(v)
0.16*0.6 = 0.096
0.36*0.4 = 0.144
0.096+0.144 = 0.24
(variance) 2
= 6 - (2.4)2 = 0.24
(standard deviation)  = sqrt(0.24) = 0.4899
OR
variance by computing formula:
x*P(x)
x
P(x)
2*0.6 = 1.2
3*0.4 = 1.2
1.2+1.2 = 2.4
2
3
0.6
0.4
v-µ
2-2.4 = -0.4
3-2.4 = 0.6
(variance) 2
= 0.24
(standard deviation)  = sqrt(0.24) = 0.4899
Problem #56
µ = 1*0.3 + 2*0.4 + 3*0.3 = 2
variance by definition formula:
2 = (1-2)2*0.3 + (2-2)2*0.4 + (3-2)2*0.3 = 0.6
(standard deviation) 
= sqrt(0.6) = 0.7746
variance by computing formula:
E(y2) = 12*0.3 + 22*0.4 + 32*0.3 = 4.6
2 = 4.6 - (2)2 = 0.6
(standard deviation) 
= sqrt(0.6) = 0.7746
OR
variance by computing formula:
y*P(y)
y
1*0.3 = 0.3
1
2*0.4 = 0.8
2
3*0.3 = 0.9
3
0.3+0.8+0.9 = 2
P(y)
0.3
0.4
0.3
y2
1
4
9
y2*P(y)
1*0.3 = 0.3
4*0.4 = 1.6
9*0.3 = 2.7
0.3+1.6+2.7 = 4.6
2 = 4.6 - (2)2 = 0.6
(standard deviation)  = sqrt(0.6) = 0.7746
OR
variance by computing formula:
y*P(y)
y
P(y)
1*0.3 = 0.3
2*0.4 = 0.8
3*0.3 = 0.9
0.3+0.8+0.9 = 2
(variance) 2
1
2
3
0.3
0.4
0.3
y-µ
1-2 = -1
2-2 = 0
3-2 = 1
(y-µ)2
1
0
1
(y-µ)2*P(y)
1*0.3 = 0.3
0*0.4 = 0
1*0.3 = 0.3
0.3+0+0.3 = 0.6
= 0.6
(standard deviation) 
= sqrt(0.24) = 0.7746
Problem #57
µ = 3*0.1 + 5*0.5 + 6*0.4 = 5.2
variance by definition formula:
2 = (3-5.2)2*0.1 + (5-5.2)2*0.5 + (6-5.2)2*0.4 = 0.76
(standard deviation) 
= sqrt(0.76) = 0.8718
variance by computing formula:
E(z2) = 32*0.1 + 52*0.5 + 62*0.4 = 27.8
2 = 27.8 - (5.2)2 = 0.76
(standard deviation) 
= sqrt(0.76) = 0.8718
OR
variance by computing formula:
z*P(z)
z
3*0.1 = 0.3
3
5*0.5 = 2.5
5
6*0.4 = 2.4
6
0.3+2.5+2.4 = 5.2
P(z)
0.1
0.5
0.4
2 = 27.8 - (5.2)2 = 0.76
(standard deviation)  = sqrt(0.76) = 0.8718
OR
z2
9
25
36
z2*P(z)
9*0.1 = 0.9
25*0.5 = 12.5
36*0.4 = 14.4
0.9+12.5+14.4= 27.8
variance by computing formula:
z*P(z)
z P(z)
3*0.1 = 0.3
5*0.5 = 2.5
6*0.4 = 2.4
0.3+2.5+2.4 = 5.2
3
5
6
0.1
0.5
0.4
z-µ
3-5.2 = -2.2
5-5.2 = -0.2
6-5.2 = 0.8
(z-µ)2
4.84
0.04
0.64
(z-µ)2*P(z)
4.84*0.1 = 0.484
0.04*0.5 = 0.02
0.64*0.4 = 0.256
0.484+0.02+0.256 =
0.76
w2
0
1
w2*P(w)
0*0.4 = 0
1*0.6 = 0.6
0+0.6 = 0.6
(variance) 2
= 0.76
(standard deviation)  = sqrt(0.24) = 0.8718
Problem #58
µ = 0*0.4 + 1*0.6 = 0.6
variance by definition formula:
2 = (0-0.6)2*0.4 + (1-0.6)2*0.6 = 0.24
(standard deviation) 
= sqrt(0.24) = 0.4899
variance by computing formula:
E(w2) = 02*0.4 + 12*0.6 = 0.6
2 = 0.6 - (0.6)2 = 0.24
(standard deviation) 
= sqrt(0.24) = 0.4899
OR
variance by computing formula:
w*P(w)
w
0*0.4 = 0
0
1*0.6 = 0.6
1
0+0.6 = 0.6
(variance) 2
P(w)
0.4
0.6
= 0.6 - (0.6)2 = 0.24
(standard deviation)  = sqrt(0.24) = 0.4899
OR
variance by computing formula:
w*P(w)
w
P(w)
0*0.4 = 0
1*0.6 = 0.6
0+0.6 = 0.6
0
1
0.4
0.6
w-µ
0-0.6 = -0.6
1-0.6 = 0.4
(w-µ)2
0.36
0.16
(w-µ)2*P(w)
0.36*0.4 = 0.144
0.16*0.6 = 0.196
0.144+0.096 = 0.24
y2
4
4
y2*P(y)
4*0.6 = 2.4
4*0.4 = 1.6
2.4+1.6 = 4
(y-µ)2
2.56
5.76
(y-µ)2*P(y)
2.56*0.6 = 1.536
5.76*0.4 = 2.304
1.536+2.304 = 3.84
(variance) 2
= 0.24
(standard deviation) 2 = sqrt(0.24) = 0.4899
Problem #59
µ = -2*0.6 + 2*0.4 = -0.4
variance by definition formula:
2 = (-2-(-0.4))2*0.6 + (2-(-0.4))2*0.4 = 3.84
(standard deviation)  = sqrt(3.84) = 1.9596
variance by computing formula:
E(y2) = (-2)2*0.6 + 22*0.4 = 4
2 = 4 - (-0.4)2 = 3.84
(standard deviation)  = sqrt(3.84) = 1.9596
OR
variance by computing formula:
y*P(y)
y
-2*0.6 = -1.2
-2
2*0.4 = 0.8
2
-1.2+0.8 = -0.4
P(y)
0.6
0.4
(variance) 2
= 4 - (-0.4)2 = 3.84
(standard deviation)  = sqrt(3.84) = 1.9596
OR
variance by computing formula:
y*P(y)
y P(y)
-2*0.6 = -1.2
2*0.4 = 0.8
-1.2+0.8 = -0.4
-2
2
0.6
0.4
y-µ
-2-(-0.4) = -1.6
2-(-0.4) = 2.4
(variance) 2
= 3.84
(standard deviation)  = sqrt(3.84) = 1.9596
Problem #60
µ = -2*0.5 + 2*0.5 = 0
variance by definition formula:
2 = (-2-0)2*0.5 + (2-0)2*0.5 = 4
(standard deviation)  = sqrt(4) = 2
variance by computing formula:
E(y2) = (-2)2*0.5 + 22*0.5 = 4
2 = 4 - 02 = 4
(standard deviation) 
= sqrt(4) = 2
OR
variance by computing formula:
y*P(y)
y
-2*0.5 = -1
-2
2*0.5 = 1
2
-1+1 = 0
P(y)
0.5
0.5
y2
4
4
y2*P(y)
4*0.5 = 2
4*0.5 = 2
2+2 = 4
(y-µ)2
4
4
(y-µ)2*P(y)
4*0.5 = 2
4*0.5 = 2
2+2 = 4
(variance) 2
= 4 - 02 = 4
(standard deviation)  = sqrt(4) = 2
OR
variance by computing formula:
y*P(y)
y P(y)
-2*0.5 = -1
2*0.5 = 1
-1+1 = 0
(variance) 2
-2
2
0.5
0.5
y-µ
-2-0 = -2
2-0 = 2
=4
(standard deviation) 
= sqrt(4) = 2
Problem #61
µ = 0*0.5 + 4*0.3 + 6*0.2 = 2.4
variance by definition formula:
2 = (0-2.4)2*0.5 + (4-2.4)2*0.3 + (6-2.4)2*0.2 = 6.24
(standard deviation) 
= sqrt(6.24) = 2.4980
variance by computing formula:
E(w2) = 02*0.5 + 42*0.3 + 62*0.2 = 12
2 = 12 - (2.4)2 = 6.24
(standard deviation) 
= sqrt(6.24) = 2.4980
OR
variance by computing formula:
w*P(w)
w
0*0.5 = 0
0
4*0.3 = 1.2
4
6*0.2 = 1.2
6
0+1.2+1.2 = 2.4
P(w)
0.5
0.3
0.2
w2
0
16
36
w2*P(w)
0*0.5 = 0
16*0.3 = 4.8
36*0.2 = 7.2
0+4.8+7.2= 12
2 = 12 - (2.4)2 = 6.24
(standard deviation)  = sqrt(6.24) = 2.4980
OR
variance by computing formula:
w*P(w)
w P(w)
0*0.5 = 0
4*0.3 = 1.2
6*0.2 = 1.2
0+1.2+1.2 = 2.4
0
4
6
0.5
0.3
0.2
(variance) 2
w-µ
0-2.4 = -2.4
4-2.4 = 1.6
6-2.4 = 3.6
= 6.24
(standard deviation)  = sqrt(6.24) = 2.4980
Problem #62
µ = 2*0.1 + 6*0.5 + 10*0.4 = 7.2
(w-µ)2
5.76
2.56
12.96
(w-µ)2*P(w)
5.76*0.5 = 2.88
2.56*0.3 = 0.768
12.96*0.2 = 2.592
2.88+0.768+2.592 =
6.24
variance by definition formula:
2 = (2-7.2)2*0.1 + (6-7.2)2*0.5 + (10-7.2)2*0.3 = 6.56
(standard deviation) 
= sqrt(6.56) = 2.5612
variance by computing formula:
E(x2) = 22*0.1 + 62*0.5 + 102*0.3 = 58.4
2 = 58.4 - (7.2)2 = 6.56
(standard deviation) 
= sqrt(6.56) = 2.5612
OR
variance by computing formula:
x*P(x)
x
2*0.1 = 0.2
2
6*0.5 = 3
6
10*0.4 = 4
10
0.2+3+4 = 7.2
P(x)
0.1
0.5
0.4
x2
4
36
100
x2*P(x)
4*0.1 = 0.4
36*0.5 = 18
100*0.4 = 40
0.4+18+40= 58.4
2 = 58.4 - (7.2)2 = 6.56
(standard deviation)  = sqrt(6.56) = 2.5612
OR
variance by computing formula:
x*P(x)
x P(x)
2*0.1 = 0.2
6*0.5 = 3
10*0.4 = 4
0.2+3+4 = 7.2
(variance) 2
2
6
10
0.1
0.5
0.4
x-µ
2-7.2 = -5.2
6-7.2 = -1.2
10-7.2 = 2.8
= 6.56
(standard deviation)  = sqrt(6.56) = 2.5612
(x-µ)2
27.04
1.44
7.84
(x-µ)2*P(x)
27.04*0.1 = 2.704
1.44*0.5 = 0.72
7.84*0.4 = 3.136
2.704+0.768+3.136
= 6.56
Problem #63(a)
µ = 0*0.20+1*0.25+2*0.30 + 3*0.15 + 4*0.10 = 1.7
OR
x*P(x)
x
P(x)
0*0.20 = 0
0
0.20
1*0.25 = 0.25
1
0.25
2*0.30 = 0.60
2
0.30
3*0.15 = 0.45
3
0.15
4*0.1 = 0.40
4
0.10
0+0.25+0.60+
0.45+0.40 = 1.70
Problem #63(b)
variance by definition formula:
2 = (0-1.7)2*0.20 + (1-1.7)2*0.25 + (2-1.7)2*0.30 + (3-1.7)2*0.15
+ (4-1.7)2*0.10 = 1.51
variance by computing formula:
E(x2) = 02*0.20 + 12*0.25 + 22*0.30 + 32*0.15 + 42*0.10 = 4.4
2 = 4.4 - (1.7)2 = 1.51
OR
variance by computing formula:
x*P(x)
x
0*0.20 = 0
0
1*0.25 = 0.25
1
2*0.30 = 0.60
2
3*0.15 = 0.45
3
4*0.1 = 0.40
4
0+0.25+0.60+
0.45+0.40 = 1.70
2 = 4.4 - (1.7)2 = 1.51
OR
P(x)
0.20
0.25
0.30
0.15
0.10
x2
0
1
4
9
16
x2*P(x)
0*0.20 = 0
1*0.25 = 0.25
4*0.30 = 1.20
9*0.15 = 1.35
16*0.10 = 1.60
0+0.25+1.20+
1.35+1.60 = 4.4
variance by computing formula:
x*P(x)
x P(x)
0*0.20 = 0
1*0.25 = 0.25
2*0.30 = 0.60
3*0.15 = 0.45
4*0.1 = 0.40
0+0.25+0.60+
0.45+0.40 = 1.70
(variance) 2
0
1
2
3
4
0.20
0.25
0.30
0.15
0.10
x-µ
0-1.7 = -1.7
1-1.7 = -0.7
2-1.7 = 0.3
3-1.7 = 1.3
4-1.7 = 2.3
(x-µ)2
2.89
0.49
0.09
1.69
5.29
(x-µ)2*P(x)
2.89*0.20 = 0.578
0.49*0.25 = 0.1225
0.09*0.3 = 0.027
1.69*0.15 = 0.2535
5.29*0.10 = 0.529
0.578+0.1225+0.027
+0.2535+0.529 = 1.51
= 1.51
Problem #63(c)
(standard deviation) 
= sqrt(1.51) = 1.2288
Problem #64(a)
µ = (-3)*0.1 + (-1)*0.15 + 0*0.5 + 1(0.15) + 3(0.1) = 0
OR
x*P(x)
x
P(x)
-3*0.10 = -0.3
-3
0.10
-1*0.15 = -0.15
-1
0.15
0*0.50 = 0
0
0.50
1*0.15 = 0.15
1
0.15
3*0.1 = 0.30
3
0.10
-0.3 + (-0.15)+ 0 +
0.15 + 0.30 = 0
variance by definition formula:
2 = (-3-0)2*0.1 + (-1-0)2*0.15 + (0-0)2*0.5+ (1-0)2*0.15 + (3-0)2*0.1 =
2.1
(standard deviation)  = sqrt(2.1) = 1.4491
variance by computing formula:
E(x2) = (-3)2*0.1 + (-1)2*0.15 + 02*0.5+ (1)2*0.15 + 32*0.1 = 2.1
2 = 2.1 - (0)2 = 2.1
(standard deviation) 
OR
= sqrt(2.1) = 1.4491
variance by computing formula:
x*P(x)
x
-3*0.1 = -0.3
-3
-1*0.15 = -0.15
-1
0*0.5 = 0
0
1*0.15 = 0.15
1
3*0.1 = 0.3
3
-0.3+(-0.1)+0+
0.15+0.3 = 0
P(x)
0.1
0.15
0.5
0.15
0.1
x2
9
1
0
1
9
x2*P(x)
9*0.1 = 0.9
1*0.15 = 0.15
0*0.5 = 0
1*0.15 = 0.15
9*0.1 = 0.9
0.9+0.15+0+
0.15+0.9= 2.1
2 = 2.1 - (0)2 = 2.1
(standard deviation)  = sqrt(2.1) = 1.4491
OR
variance by computing formula:
x*P(x)
x P(x)
-3*0.1 = -0.3
-1*0.15 = -0.15
0*0.5 = 0
1*0.15 = 0.15
3*0.1 = 0.3
-0.3+(-0.1)+0+
0.15+0.3 = 0
(variance) 2
-3
-1
0
1
3
0.1
0.15
0.5
0.15
0.1
x-µ
-3-0 = -3
-1-0 = -1
0-0 = 0
1-0 = 1
3-0 = 3
= 2.1
(standard deviation) 
= sqrt(2.1) = 1.4491
(x-µ)2
9
1
0
1
9
(x-µ)2*P(x)
9*0.1 = 0.9
1*0.15 = 0.15
0*0.5 = 0
1*0.15 = 0.15
9*0.1 = 0.9
0.9+0.15+0+
0.15+0.9= 2.1
Problem #64(b)
Since  = sqrt(2.1) = 1.4491, then 2
= 2*1.4491 = 2.8982
So µ-2 = 0 – 2.8982 = -2.8982 and
So µ+2 = 0 + 2.8982 = 2.8982 so the histogram looks like:
Problem 64
0.6
µ-2
µ+2
µ
0.5
0.4
0.3
0.2
0.1
0
-3
-1
0
1
3
Problem #64(c)
Since the probability must be between +/-2.8982, then the probability cannot
include the probability of 3 or -3…so the answer is 0.15+0.5+0.15 = 0.8
Problem #65
Both means will be at 3 so that the probabilities will “balance” at that
point (equal amounts on both sides of 3).
Problem #66
Y since more of the probability is at the mean (less of a standard
deviation, or spread in the data).