Environmental Engineering Reference
In-Depth Information
a) b)
1. 13 Distribution f ( t ).
is a criterion for testing the hypothesis of the random variable T , represented
by its sample, has the distribution of the expected type.
The following procedure is used for verification. The criterion
is calculated as a measure of the discrepancy between the theoretical
and empirical distributions, and this measure is a random variable. The
higher the measure of discrepancy, the larger the difference between the
empirical and theoretical distributions, i.e. the hypothesis for the choice
of the distribution should be rejected as highly unlikely. Otherwise the
experimental data do not contradict the accepted distribution.
Of the known criteria, the Pearson criterion χ 2 (chi-square) is used most
widely. The consistency of distributions using the χ 2 criterion is verified as:
- criterion χ 2 is calculated (a measure of divergence)
2
ˆ
K
(
PP
)
c=
2
N
i
i
,
P
1
i
￿ ￿ ￿ ￿ ￿
ˆ ()
where
is the theoretical frequency (probability) of getting a
random variable in the interval [ t i , t i + t ];
- the number of degrees of freedom is determined R = k - L , where L is
the number of independent conditions imposed on frequency P i , for example:
a) condition;
iP ft t
=
D
i
i
ˆ
S= ;
b) the condition of coincidence;
1
ˆ
ˆ
S=
tP T
;
ii
0
c) the condition of coincidence (
) 2
S= , etc.
In most cases, L = 3. The greater the number of degrees of freedom, the
greater the random variable χ 2 that obeys the Pearson distribution;
- the calculated χ 2 and R are used to determine the probability P that the
value having the Pearson distribution with R degrees of freedom exceeds
the calculated value of χ 2 .
The answer to the question: how small must probability P be to reject
the hypothesis that the choice of a distribution law is largely undefined.
ˆ
ˆ
ˆ i
t TPD
0
i
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