https:inst.eecs.berkeley.edu:~ee126:sp20:hw:hw2-sol.pdf -...

UC Berkeley
Department of Electrical Engineering and Computer Sciences
EECS 126: Probability and Random Processes
Problem Set 2
Spring 2020
1 . Variance
If X
1
, . . . , X
n
, where n 2 Z
> 0
, are i.i.d. random variables with zero-mean and
unit variance, compute the variance of ( X
1
+ · · · + X
n
)
2
. You may leave your
answer in terms of E [ X
4
1
], which is assumed to be finite.
2 . Message Segmentation
The number of bytes N in a message has a geometric distribution with parameter
p . Suppose that the message is segmented into packets, with each packet
containing m bytes if possible, and any remaining bytes being put in the last
packet. Let Q denote the number of full packets in the message, and let R
denote the nubmer of bytes left over.
1

(a) Find the joint PMF of Q and R . Pay attention on the support of the
joint PMF.
(b) Find the marginal PMFs of Q and R .
(c) Repeat part (b), given that we know that N > m .
Note: you can use the formulas
n X
k =0
a
k
=
1 - a
n +1
1 - a
, for a 6 = 1
1
X
k =0
x
k
=
1
1 - x
, for | x | < 1
in order to simplify your answer.
2

= p (1 - p )
m - 1
1
X
q =0
(
(1 - p )
m
)
q
=
p (1 - p )
m - 1
1 - (1 - p ) m
.
(c) Due to the memoryless property of the geometric distribution, the PMF