Fundamentals of Probability and Statistics for Engineers

However,

H ence,

where

is a constant.
The desired fR(r A) is then obtained from the above by differentiation. We
obtain

It can be seen from Figure 3.18 (dashed line) that the effect of screening is
essentially a truncation of the tails of the distribution beyond the allowable
limits. This is accompanied by an adjustment within the limits by a multi-
plicative factor 1/c so that the area under the curve is again equal to 1.

Further Reading and Comments

We discussed in Section 3.3 the determination of (unique) marginal distributions from a
knowledge of joint distributions. It should be noted here that the knowledge of marginal
distributions does not in general lead to a unique joint distribution. The following reference
shows that all joint distributions having a specified set of marginals can be obtained by
repeated applications of the so-called transformation to the product of the marginals:

Becker, P.W., 1970, ‘‘A N ote on Joint D ensities which have the Same Set of M arginal
Densities’’, in Proc. International Symp. InformationTheory ,Elsevier Scientific Pub-
lishers, The Netherlands.

66 Fundamentals of Probability and Statistics for Engineers

Rr\ 48 R 52 ˆ

;; forr< 48 ;

48 Rr; for 48r 52 ;

48 R 52 ; forr> 52 :

8

><

>:

FR…rjA†ˆ

0 ; forr< 48 ;

P… 48 Rr†

P… 48 R 52 †

ˆ

Zr

48

fR…r†dr

c

; for 48r 52 ;

1 ; forr> 52 ;

8

>>

>>

><

>>>

>>

:

cˆ

Z 52

48

fR…r†dr:

j

fR…rjA†ˆ

dFR…rjA†

dr

ˆ

fR…r†

c

; for 48r 52

0 ;elsewhere

8

<

: