# Homework Solutions

##
ECE 6010 Stochastic Processes

Homework #3

Problems from Grimmet & Stirzaker:

1. Prob 2.7.4

a)

b)

c)

d)

e)

f)

if

2.Prob 2.7.7.
Let T be the numbers of people on given typical flights of TWA,
and B be the numbers of people on given typical flights of BA.

Now P(TWA overbooked)= P(T=10)=

P(BA overbooked) of which the latter is the larger.

3. Prob 2.7.9.
(a)

(b)

(c)
, if
.
Therefore.

(d)

4. Ex 3.3.1.
(a) No!

(b) Let X have mass function:
Then

5. Ex 3.4.1.
Let
be the indicator function of the event that the
outcome of the (
*
i
*
+1)th toss is different from the outcome of
the
*
j
*
th toss. The number
*
R
*
of distinct runs is
given by
. Hence

where . Now remark that and are independent if , so that

Now and , and therefore

6. Ex 3.4.2. The required total is
,
where
is the number shown on the
*
i
*
th ball. Hence

Hence

7. Ex 3.5.2. The total number H of heads satisfies

The last summation equals 1, since it is the sum of the values of the Poission mass function with parameter

8. Ex 3.6.5. (a)
with equality if and only if
. Therefore,

with equality if an only if

(b)

so