Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 89

2.2.2 Expectation Value of a Random Variable

The expectation value of a random variable Y is

= + ∞
−∞

E Y yp Y ( y ) dy.

Therefore, the expectation value of a random variable is the first moment
of its probability distribution .

2.2.2.1 Properties

• The expectation value of the sum of random variables is the sum of the

expectation values of the random variables.

• If a variable Y is uniformly distributed in interval [ a , b ], its expectation

value is ( a + b ) / 2.

• If a variable Y has a Gaussian distribution with mean µ , its expectation

value is µ .

2.2.2.2 Example: Modeling the Result of a Measurement by a
Random Variable

Assume that several measurements of the temperature of a fluid are per-
formed, under conditions that are assumed to be identical, and that different
results are obtained because of the intrinsic noise of the sensor and associ-
ated electronics, or because the conditions of the measurement are poorly
controlled. Such a situation can be conveniently modeled by considering that
the result T of the measurement is the sum of the true temperature T 0 (ran-
dom variable with distribution δ ( T 0 )) and of a random variable B with zero
expectation value, T = T 0 + B . Then the expectation value of T is given by
E T = T 0 since the expectation value of B is equal to zero.
Clearly, the objective of performing a measurement of a quantity of inter-
est, is to know its “true” value, i.e., within the above statistical framework,
the expectation value of the quantity of interest. Therefore, the question that
arises naturally is: how can one estimate that expectation value from the
available measurements? To this end, the concept of estimator is useful.

2.2.3 Unbiased Estimator of a Parameter of a Distribution

An estimator is a random variable, which is a function of one or several mea-
surable random variables.
An estimator H of a parameter of the distribution of an observable ran-
dom variable G is said to be unbiased if its expectation value is equal to the
parameter of interest. Then a realization of H is an unbiased estimate of the
parameter of interest.