Page 83 - Fundamentals of Probability and Statistics for Engineers

66 Fundamentals of Probability and Statistics for Engineers

However,
;; for r < 48;
8
>
<
R r \ 48 R 52 ˆ 48 R r; for 48 r 52;
>
48 R 52; for r > 52:
:
Hence,
0; for r < 48;
8
>
>
> Z r
>
f r†dr
>
<
R
F R rjA†ˆ P 48 R r† 48
ˆ ; for 48 r 52;
P 48 R 52† c
>
>
>
>
>
:
1; for r > 52;
where
52
Z
c ˆ f r†dr:
R
48
is a constant.
The desired f (r A) is then obtained from the above by differentiation. Wej
R
obtain
f R r†
8
dF R rjA† < ; for 48 r 52
f rjA†ˆ ˆ c
R
dr :
0; elsewhere
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 Note on Joint Densities which have the Same Set of Marginal

Densities’’, in Proc. International Symp. Information Theory, Elsevier Scientific Pub-
lishers, The Netherlands.

TLFeBOOK