Stat 309
HW7-Chapter 3
1/5
Home Work 7
16
.What is the probability density of the time between the arrival of the two packets of Example E in
Section 3.4?
Ans:
Let
X
=
∣
T
1
−
T
2
∣
Then
F
X
x
=
P
X
x
=
P
∣
T
1
−
T
2
∣
x
=
P
−
x
T
1
−
T
2
x
=
∬
A
f
T
1,
T
2
t
1,
t
2
dt
1
dt
2
with
A is the
area when
−
x
T
1
−
T
2
x
, the shaded strip in Figure 3.12. From this example we
have area of A is
T
2
−
T
−
x
2
.
Since
T
1
,T
2
are independent uniform random variables on [1,T] we have:
f
T
1
t
1
=
1
T
,
∀
t
1
∈[
0,
T
]
and 0 otherwise,
f
T
2
t
2
=
1
T
,
∀
t
2
∈[
0,
T
]
and 0 otherwise,
f
T
1
,T
2
t
1
,t
2
=
f
T
1
t
1
f
T
1
t
1
=
1
T
2
,
∀
t
1
∈[
0,
T
]
,t
2
∈[
0,
T
]
and 0 otherwise.
Hence:
F
X
x
=
∬
A
1
T
2
dt
1
dt
2
=
1
T
2
∬
A
dt
1
dt
2
=
1
T
2
∗
Area of region
A
=
T
2
−
T
−
x
2
T
2
=
1
−
1
−
x
T
2
So probability density of X is
f
X
x
=
d
dx
F
X
x
=
d
dx
1
−
1
−
x
T
2
=
2
T
1
−
x
T
24.
Let P have a uniform distribution on [0,1], and, conditional on P=p, let X have a Bernulli
distribution with parameter p. Find the conditional distribution of P given X.
Ans:
By definition of conditional density we have conditional density of P given X is:
f
P
∣
X
p
∣
x
=
f
X , P
x , p
f
X
x
So we need to find joint density X, P and marginal density of X.
From question we have:
●
P have a uniform distribution on [0,1] so density (pdf) of P is:
f
P
p
=
1
,
∀
p
∈[
0,1
]
and 0 otherwise.
●
Conditional on P=p, X has a Bernulli distribution with parameter p so conditional pdf of X
given P is:
f
X
∣
P
x
∣
p
=
p
x
1
−
p
1
−
x
,if x
=
0
∨
x
=
1
Hence by multiplicative law we have joint pdf of X, P:
f
X , P
x , p
=
f
X
∣
P
x
∣
p
f
P
p
=
p
x
1
−
p
1
−
x
,
if x
=
0
∨
x
=
1,
p
∈[
0,1
]
and 0 otherwise.
By definition of marginal density we have:
Quoc Tran
B248D MSC
tran@stat.wisc.edu