Page 301 - DCOM303_DMGT504_OPERATION_RESEARCH

Operations Research

Notes 14.4.5 Criterion of Realism (Hurwicz Criterion)

This criterion is a compromise between an optimistic and pessimistic decision criterion. To start
with, a coefficient of optimism a (0  a  1), is selected. When a is close to one, the decision maker
is optimistic about the future, and when a is close to zero the decision maker is pessimistic about
the future. According to Hurwicz, select strategy which maximizes:

H = a (Maximum payoff in column) + (1 – a) (Minimum payoff in column)

Task Describe a business situation where a decision maker faces a decision-making
under uncertainty and where a decision based on maximizing the expected monetary
value cannot be made. How do you think the decision maker should make the required
decision?

14.5 Expected Value

The expected value (or expectation value, or mathematical expectation, or mean, or first moment)
of a random variable is the integral of the random variable with respect to its probability
measure. For discrete random variables this is equivalent to the probability-weighted sum of
the possible values, and for continuous random variables with a density function it is the
probability density-weighted integral of the possible values.

!
Caution The term "expected value" can be misleading. It must not be confused with the
"most probable value." The expected value is in general not a typical value that the random
variable can take on. It is often helpful to interpret the expected value of a random variable
as the long-run average value of the variable over many independent repetitions of an
experiment.
The expected value may be intuitively understood by the law of large numbers: The expected
value, when it exists, is almost surely the limit of the sample mean as sample size grows to
infinity. The value may not be expected in the general sense - the "expected value" itself may be
unlikely or even impossible (such as having 2.5 children), just like the sample mean. The expected
value does not exist for all distributions, such as the Cauchy distribution.

It is possible to construct an expected value equal to the probability of an event by taking the
expectation of an indicator function that is one if the event has occurred and zero otherwise. This
relationship can be used to translate properties of expected values into properties of probabilities,
e.g. using the law of large numbers to justify estimating probabilities by frequencies.
In general, if X is a random variable defined on a probability space (, , P), then the expected
value of X, denoted E(X), (X), X or E(X), is defined as

E(X) = X dP
 
where the Lebesgue integral is employed. Note that not all random variables have an expected
value, since the integral may not exist. Two variables with the same probability distribution
will have the same expected value, if it is defined.
If X is a discrete random variable with probability mass function p(x), then the expected value
becomes E(X)   x p(x ) as in the gambling example mentioned above.
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