Page 173 - Artificial Intelligence for the Internet of Everything
P. 173
The Value of Information and the Internet of Things 159
E (V | b)
0.5
0.4
0.3
0.2
0.1
b
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Fig. 9.5 EðVjbÞ 0 when the company knows the cost, L¼ U½0,2 and PðC ¼ 1=2Þ¼ 1.
b 1=2 PðL > bÞ, if 1=2 < b 2
(9.21)
½
EðVjbÞ¼
0, otherwise
Eq. (9.21) simplifies to:
8
2 b
<
, if 1=2 < b 2
ð b 1=2Þ (9.22)
2
EðVjbÞ¼
0, otherwise
:
In Example 9.1, EðVjbÞ has a maximum value of 18/32, when b ¼ 5/4.
Example 9.2 [L¼ U½0,2 and C¼ U½0,1] Eq. (9.20) now becomes:
8
Z b
b 0 1 2 b
>
> , if 0 b 1
b c dc
>
1 0 1 2
>
< 0
Z 1 (9.23)
EðVjbÞ¼ 2 b
, if 1 < b 2
>
b PðC 1Þ cdc
>
∞
> 2
>
:
0, otherwise
Eq. (9.23) simplifies to:
b
8 2
ð2 bÞ,
>
> b 2½0,1
4
>
<
2 b 1 1
EðVjbÞ¼
,
½ b EðCÞ ¼ ð2 bÞ b b 2ð1,2
>
2 2 2
>
>
:
0, otherwise
(9.24)
