RANDOM PROCESSES
Homework Problems Solutions
1. (5 points) A register contains 10 random binary digits, which are mutually
independent. Each digit is “0” or “1” with equal probability. There is an
appropriate probability space (S,F,P) corresponding to the contents of the
register.
(a) Express explicitly an event consisting of a run of at least nine
consecutive ”1”s and find its probability.
(b) Express explicitly the event E2 consisting of a run of exactly nine (not
only consecutive) ”1”s and find its probability.
(c) Find P{E1|E2].
(a)
(b)
(c)
01111111111, 1111111110, 1111111111, P =3/210
11111111111, 0111111111, 1011111111, 1101111111,…. 1111111110,
P =10/210
P = 2/10
2. (5 points) The probability density of a gas molecule having velocity v is
f V (v)  Ae
2
 mv
2 kT
v 2 , where A, m, k, and T are constants. Find the probability
density function for the molecule kinetic energy E  mv 2 / 2 .
f v (v)dv  f E ( E )dE
f E ( E )  f v v( E ) dvdE( E )  Ae
E
 kT
2E
m
 m2 1/ 2 12 E 1/ 2  Ae
E
kT
21 / 2
m3 / 2
E1/ 2
3. ((5 points) Let X and Y be defined as:
X  cos  ,
Y  sin 
where  is a random variable uniformly distributed over (0,2).
(a) Show that X and Y are uncorrelated.
(b) Show that X and Y are not independent.
Look at the problem 4.42 in Schaum’s Outlines book on Random Processes
.
4. (10 points) Prove the following two properties of the autocorrelation function
R X ( ) of a random process X(t):
(a) If X(t) contains a DC component equal to A, than R X ( ) will also
contain a constant component equal to A2.
(b) If X(t) contains a sinusoidal component equal to A, than R X ( ) will
also contain a sinusoidal component with the same frequency.
Solutions follows directly from the relation defining R X ( ) .
5. (5 points) Prove that the power spectral density of a real-valued random
process is an even function of frequency.

R
S X ( ) 
X
( )e
j
d 


R
X
( )e
 j
d 


R
X
( )e  j d  S X ( )

6. (10 points) The output of an oscillator is given by X (t )  A cos( 2ft   ) ,
where A is a constant, and f and  are independent random variables. The
probability density function of  is defined by:
 1 0    2
f  ( )   2 ,
 0, otherwise
Find the power spectral density of X(t) in terms of the probability density
function of the frequency f.
Rx    E X (t   ) X (t )

A2
2

A2
2
E cos4ft  2f  2  
A2
2
E cos2f 
  2
  cos4ft  2f  2  f ( ) f
 0

( f )ddf 
 2

1


cos
4

ft

2

f


2

d

 f f ( f )df 
2
  0

 cos2f  f
f
( f )df
f
( f )df

0

A2
2
f

A2
2

A2
2
 cos2f  f



A2
2
 cos2f  f
f
( f )df


S X ( )   R X ( )e

 j
d 

  j


cos
2

f

f
(
f
)
df
e d
f


 
A2
2
If f assumes constant value fc, than:
S X ( ) 

  jc
2


cos


f
(
f
)
df
d  A2  (  c )   (  c )
e
c  f




2
A
2
7. (5 points) Find the autocorrelation function R X ( ) and autocovariance
function K X ( ) of a Poisson process with rate .
EX (t )  t , VarX (t )  t
K X (t , s)   min( t, s) , RX (t , s)   min( t , s)  2ts
Look at the problem 5.52 in Schaum’s Outlines book on Random Processes.
USEFUL FORMULAS FOR:
All necessary distribution, all necessary distribution
moments, all necessary Fourier transforms and Wiener –
Khinchin will be available for the test.
I will be available for consultations on Sunday, February
19, between 12:00 and 13:00, in my office.