STATS 218: Stochastic Processes
Homework 1 Solution
1. 1.11
(1) We can use the fact that, on its radius of convergence, a power series can be dierentiated term by
term. The radius of convergence includes cfw_z : |z| 1, since, from the problem statement, P
STATS 218: Stochastic Processes
Homework 2 Solution
2.30
(a). Note that
r+t
P (T2 > t | T1 = r) = exp
(s) ds ,
r
which for general (s) general depends on r, so T1 and T2 are not independent.
(b). In parts (c) and (d), we will see that the densities of T
STATS 218: Stochastic Processes
Homework 3 Solution
3.8
(a). We can write the distribution function as
P (X1 x1 , . . . , Xn xn | N (t) = n) =
P (X1 x1 , . . . , Xn xn , N (t) = n)
P (N (t) = n)
xn
=
=
.
.
.
x1
n
P (Xn+1 > t i=1 zi ) [
n
P (Xn+1 > t i=1 z
STATS 218: Stochastic Processes
Homework 4 Solution
3.4
We condition on the time of the rst arrival:
E [N (t)] = E [E [1 (x1 t) (N (t X1 ) + 1) | X1 ]
t
(N (t s) + 1) dF (s)
=
0
t
m (t s) dF (s) .
= F (t) +
0
3.5
The renewal equation can be written as
m =