Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 91

2.2.4.1 Properties

E 2
Y

• var Y = E Y 2 − .

= a 2 var Y .

• var aY

• If a random variable is uniformly distributed on an interval [ a,b ], its vari-

ance is ( b

a ) 2 / 12.

−

• If a random variable has a Gaussian distribution of standard deviation σ ,

its variance is σ 2 .

2.2.4.2 Unbiased Estimator of the Variance of a Random Variable

In order to define the mean estimator M (unbiased estimator of the expec-
tation value), we considered that N measurements of a quantity G were per-
formed, and that the measurements were modeled as realizations of N inde-
pendent identically distributed (i.i.d.) random variables G i .

Unbiased Estimator of the Variance

The random variable

N

1

S 2 = ( G i −M ) 2

N − 1

i =1

is an unbiased estimator of the variance of G .
Therefore, if N measurement results g i are available, the estimation of the
variance requires

first an estimation m of the mean, by relation m =1 / ( N ) i =1 g i ,

•

• then an estimation of the variance by relation

N

1

s 2 = m ) 2 . ( g i −

N − 1

i =1

Thus, the estimation of the variance provides a quantitative assessment of
the scattering of the measurements around the mean. Since the mean itself is
a random variable, it has a variance: the latter can be estimated by perform-
ing several sequences of measurements, under identical conditions, computing
the mean of each sequence, then estimating the expectation value and the
variance of the mean: this would provide an assessment of the scattering of
the estimates of the temperature. However, this is indeed a heavy procedure,
since it requires several sequences of measurements, in identical conditions.