Uncertainty Theory
Baoding Liu @ Tsinghua University
1
Expected
Value
Liu
Let ξ be an uncertain variable. Then the expected value of ξ is defined by
Z +∞
Z 0
E [ξ] =
M{ξ ≥ r }dr −
M{ξ ≤ r }dr
−∞
0
provided that at least one of the two integrals is finite.
E [ξ] =
X
r >0
1
M{ξ ≥ r }∆r −
X
M{ξ ≤ r }∆r
r <0
Liu B, Uncertainty Theory, 2nd edn, Springer-Verlag, Berlin, 2007.
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Expected Value via Uncertainty Distribution - 1
Theorem
Let ξ be an uncertain variable with uncertainty distribution Φ. Then
Z +∞
Z 0
E [ξ] =
(1 − Φ(x))dx −
Φ(x)dx.
−∞
0
M{ξ ≥ x} = 1 − Φ(x)
M{ξ ≤ x} = Φ(x)
Z
E [ξ] =
+∞
M{ξ ≥ x}dx −
0
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Z
0
M{ξ ≤ x}dx
−∞
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Expected Value via Uncertainty Distribution - 2
Theorem
Let ξ be an uncertain variable with uncertainty distribution Φ. Then
Z +∞
E [ξ] =
xdΦ(x).
−∞
It follows from the integration by parts that the expected value is
Z +∞
Z 0
E [ξ] =
(1 − Φ(x))dx −
Φ(x)dx
−∞
0
Z
=
Z
0
xdΦ(x) +
0
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+∞
Z
+∞
xdΦ(x) =
−∞
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xdΦ(x).
−∞
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Expected Value via Inverse Uncertainty Distribution
Theorem
Let ξ be an uncertain variable with regular uncertainty distribution Φ.
Then
Z 1
E [ξ] =
Φ−1 (α)dα.
0
Substituting Φ(x) with α and x with Φ−1 (α), it follows from the change
of variables of integral that the expected value is
Z +∞
Z 1
E [ξ] =
xdΦ(x) =
Φ−1 (α)dα.
−∞
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Summary: Three Formulas
Z
+∞
Z
0
(1 − Φ(x))dx −
E [ξ] =
Φ(x)dx
−∞
0
Z
+∞
xdΦ(x)
E [ξ] =
−∞
Z
E [ξ] =
1
Φ−1 (α)dα
0
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Linear Uncertain Variable
The linear uncertain variable ξ ∼ L(a, b) has an expected value
E [ξ] =
Z
E [ξ] =
−1
Φ
0
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1
Z
a+b
.
2
1
((1 − α)a + αb)dα =
(α)dα =
0
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a+b
.
2
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Zigzag Uncertain Variable
The zigzag uncertain variable ξ ∼ Z(a, b, c) has an expected value
E [ξ] =
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a + 2b + c
.
4
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Normal Uncertain Variable
The normal uncertain variable ξ ∼ N (e, σ) has expected value e, i.e.,
E [ξ] = e.
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Lognormal Uncertain Variable
√
If σ < π/ 3, then the lognormal uncertain variable ξ ∼ LOGN (e, σ) has
an expected value
√
√
E [ξ] = 3σ exp(e) csc( 3σ).
Otherwise, E [ξ] = +∞.
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Expected Value of Function (Liu and Ha2 , 2010)
Theorem
Assume ξ1 , ξ2 , · · · , ξn are independent uncertain variables with regular
uncertainty distributions Φ1 , Φ2 , · · · , Φn , respectively. If the function
f (x1 , x2 , · · · , xn ) is strictly increasing with respect to x1 , x2 , · · · , xm and
strictly decreasing with xm+1 , xm+2 , · · · , xn , then
ξ = f (ξ1 , · · · , ξm , ξm+1 , · · · , ξn ) has an expected value
Z
E [ξ] =
0
1
−1
−1
−1
f (Φ−1
1 (α), · · · , Φm (α), Φm+1 (1 − α), · · · , Φn (1 − α))dα.
The inverse uncertainty distribution of ξ is
−1
−1
−1
Ψ−1 (α) = f (Φ−1
1 (α), · · · , Φm (α), Φm+1 (1 − α), · · · , Φn (1 − α)).
2
Liu YH, and Ha MH, Expected value of function of uncertain variables, Journal
of Uncertain Systems, Vol.4, No.3, 181-186, 2010.
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Uncertainty Theory
Baoding Liu @ Tsinghua University
An Example
Let ξ and η be independent and positive uncertain variables with regular
uncertainty distributions Φ and Ψ, respectively. Then
Z 1
E [ξη] =
Φ−1 (α)Ψ−1 (α)dα,
0
Z 1
ξ
Φ−1 (α)
E
=
dα.
−1
η
0 Ψ (1 − α)
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Homework
Let ξ ∼ L(1, 2) and η ∼ Z(3, 4, 7) be independent uncertain variables.
Please calculate
ξ
E
= 0.
ξ+η
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Linearity Property (Liu3 , 2010)
Theorem
Let ξ and η be independent uncertain variables with finite expected values.
Then for any real numbers a and b, we have
E [aξ + bη] = aE [ξ] + bE [η].
Step 1: If a > 0, then the inverse uncertainty distribution of aξ is aΦ−1 (α). Thus
Z 1
Z 1
E [aξ] =
aΦ−1 (α)dα = a
Φ−1 (α)dα = aE [ξ].
0
0
Step 2: By independence, the inverse uncertainty distribution of the sum ξ + η
is Υ−1 (α) = Φ−1 (α) + Ψ−1 (α). Thus
Z 1
Z 1
Z 1
E [ξ + η] =
Υ−1 (α)dα =
Φ−1 (α)dα +
Ψ−1 (α)dα = E [ξ] + E [η].
0
0
0
3
Liu B, Uncertainty Theory: A Branch of Mathematics for Modeling Human
Uncertainty, Springer-Verlag, Berlin, 2010.
[email protected]
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