I
I
4.1.6. (a) Since the interarrival times are independent and have the exp())
distribution, the
joint
c.d.f. and p.d.f. are the products of marginals:
F(s1,s2):
(1er'r)(1er'r;,
.f(sr,sz) =
()e)"')()u)"r),
sr,sz
)
0.
(b)
VrL
T1*Tz
de
*
,eW{*#
P[T, <
1, ?2
>
3]
=
P[St<1,Sr*Sz>3]
1@
lf
:
I
2""t
I
2"rt,
ds2ds1
JJ
0
3c1
I
= 
2s2""z(e"t)
dsr
J
0
:
2e6
/5rrSt=3
7
=
I
3e,e3vdu
J
o
@
r
:
e"
I
3esYdu
t
0
:

o,o3vlt6
t0
:
eo,
The conditional density can
be found by using the
definition of
f(ylx):
f
(ylr)
=
f
fr,lJ
_
3e(c+sv)
:3eBr/.
'
I"\x)
e'
The marginal density of
Y, which is
.found
by integrating the joint
den
sity over the interval
[0,
]
with respectto.r,
ulso co*es out
to
3ess.
Therefore the random variables are independent.
lr0
=
i,",,*,u,0,
u6
4.2.9. (a) We need to comPute
W+huca;t,
1
I
f(x,I/2,7,2)
f@iY=;,2:i):ffi=
for values of c
in (0,1).
4.2.6. (MM) To find the
conditional density of
y
given
c
,
first integrate
the joint
density with respect to
gr
over the interval[0, mJ
in order to find
the marginal density of a.
l(*l;,{)