Page 175 - Artificial Intelligence for the Internet of Everything
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The Value of Information and the Internet of Things 161
Modifying Eq. (9.1), now differently than Eq. (9.17), we have:
l c, l 2 support of L
(9.26)
0, otherwise
EðVjb,c,lÞ¼
We have that (C and L still independent):
Z
EðVjc,lÞ f ðcÞdc (9.27)
EðVjlÞ¼
Z
ðl cÞ f ðcÞdc (9.28)
¼
Z Z
¼ l f ðcÞdc c f ðcÞdc (9.29)
(9.30)
¼ l EðCÞ
when l 2 support of L.
Example 9.3 [L¼ U½0,L and EðCÞ ¼ 1=2] Below we show a plot of
EðVjlÞ¼ l 0:5 against l for L¼ U½0,2 and EðCÞ ¼ 1=2(Fig. 9.7).
For Example 9.3, dhVie ¼ 1:5, achieved when b ¼ 2.
b
Note that EðVjbÞ is a linear function of b ¼ l and that it can be negative,
zero (once), or positive depending on the support of L. Furthermore, the
maximum of EðVjbÞ is achieved when b is the largest value of l in the support
of L. Heuristically, another way of saying this is that the maximum is
achieved for the largest value of l such that P(L 2 (l dx, l))6¼0.
E (V | /)
1.5
1.0
0.5
l
0.5 1.0 1.5 2.0
–0.5
Fig. 9.7 EðVjbÞ such that EðCÞ ¼ 1=2, and the company knows the lowest bid,
distributed as L¼ U½0,2.
