Stochastic Programming - Index of

PROBABILISTIC CONSTRAINTS 249
4.3 Bounding Distribution Functions
In Section 4.1 we mentioned that particular methods have been developed to
compute lower and upper bounds for the function
G(x) :=P ({ξ | Tx≥ ξ}) =F˜ξ(Tx)
contained in the constraints of problem (1.1). Here F˜ξ(·) denotes the
distribution function of the random vector ˜ξ. In the following we sketch some
ideas underlying these bounding methods. For a more technical presentation,
the reader should consult the references provided below.
To simplify the notation,let us assume that ˜ξ is a random vector with a
support Ξ ⊂ IR n . For any z ∈ IR n ,wehave
F˜ξ(z) =P ({ξ | ξ 1 ≤ z 1 , ···,ξ n ≤ z n }).
Defining the events A i := {ξ | ξ i ≤ z i },i=1, ···,n, it follows that
F˜ξ(z) =P (A 1 ∩···∩A n ).
Denoting the complements of the events A i by
B i := A c i = {ξ | ξ i >z i },
we know from elementary probability theory that
A 1 ∩···∩A n =(B 1 ∪···∪B n ) c ,
and consequently
F˜ξ(z) =P (A 1 ∩···∩A n )
= P ((B 1 ∪···∪B n ) c )
=1− P (B 1 ∪···∪B n ).
Therefore asking for the value of F˜ξ(z) is equivalent to looking for the
probability that at least one of the events B 1 , ···,B n occurs. Defining the
counter ˜ν :Ξ−→ IN by
˜ν(ξ) :={number of events out of B 1 , ···,B n that occur at ξ},
˜ν is clearly a random variable having the range of integers {0, 1, ···,n}.
Observing that P (B 1 ∪···∪B n )=P (˜ν ≥ 1), we have
F˜ξ(z) =1− P (˜ν ≥ 1).
Hence finding a good approximation for P (˜ν ≥ 1) yields at the same time a
satisfactory approximation of F˜ξ(z).