# IRAS Explanatory Supplement

VIII. Sky Coverage, Confusion, Completeness and Reliability

D. Point Source Catalog Reliability and Completeness

## D.3 Estimation of Parameters

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*n*(

*N*,

*M*) of sources on the sky, observed

*N*out of

*M*possible times. There is no

*a priori*information as to whether these sources are real or false. Let

*E<x>*denote the expectation value of

*x*. Then

(VIII.D.8)

where *Q*(*N*,*M*) and *P*(*N*,*M*) are
given respectively by Eqs. (VIII.D.l) and
(VIII.D.5) and contain the parameters *p*
and *q*. It is clear that estimates *û*, *p^* and
*q^* of *u*, *p* and *q* can be obtained
by replacing the expectation value of *n*(*N*,*M*) in
Eq. (VIII.D.8) by the observed values,
bearing in mind that the value of *n*(0,*M*) is unobservable.
The following equations, from which *û, p^* and
*q^* can be estimated, are derived using Eqs.
(VIII.D.1, VIII.D5 and
VIII.D.8)

(VIII.D.9)

(VIII.D.10)

(VIII.D.11)

Since (*qA*) « 1 in the case of this survey,
Eq. (VIII.D.9) and (VIII.D.10) can
be approximated by:

(VIII.D.12)

(VIII.D.13)

The equations can be greatly simplified for sources of high signal-to-noise
ratio when *p* « 1 if it is assumed that the true source density
is not greatly in excess of the false density. Equation
(VIII.D.12) becomes

(VIII.D.14)

Equations (VIII.D.13) and (VIII.D.14) give

(VIII.D.15)

while Eqs. (VIII.D.11), (VIII.D.14) and (VIII.D.15) give

(VIII.D.16)

Each of these equations, which are valid only for bright sources, has
a simple interpretation. Equation (VIII.D.14) says
that the probability of generating false sources is given by the density
of single HCONs Pided by the number of chances of seeing them. The
contribution of true sources which have lost all but one HCON is neglected.
Equation (VIII.D.15) says that all sources with two or
more HCONs are real. Finally, Eq. (VIII.D.16) says that
the probability of missing an HCON is the sum of all the missed HCONs
divided by the total possible number of HCONs; again the contribution of
true sources which have lost *M* - 1 HCONs is neglected.

The parameters *u*, *p* and *q* were estimated using
Eqs. (VIII.D.11, 12 and 13).
Only preliminary results are presented here because these parameters were
directly affected by the last processing steps (clean-up, flux analysis,
and high source density processing) used to generate the catalog.

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