Communication Systems

Ch. 4 Probability and Random Variables

- modeling

-

- Bayes rule

- :

:

- (pdf)

Ch. 5 Random Signals and noise

-

(ensemble) : ,

- (sample function):

- (stationary process)

- (autocorrelation function) :

Ch. 6 Noise in Modulation Systems

-

-

-

- : SNR(signal to noise ratio)

- (baseband system) SNR

DSB, SSB, AM, Angle modulation, PAM, PCM, ...

Ch. 7

-

-

-

-

Ch. 8

- M(M-ary)

-

-

-

-

Chapter 4

4.1 ?

(1) (equally likely) (outcomes or sample points)

- (equally likely) :

- (mutually exclusive) :

- (chance, random experiment) N

A P(A) :

, : A

- Outcome of a random experiment is defined as a result that cannot be

decomposed into other results.

- Random experiment is experiment in which the outcome varies in an

unpredictable fashion when the experiment is repeated under the same conditions.

- The sample space of a random experiment is defined as the set S of all possible

outcomes.

- Event is the set of points from S that satisfy the given conditions. The event

occurs iff the outcome of the experiment is in this subset. Thus, eEvent is defined

as a subset of S.

- Certain event S consists of all outcomes and hence always occurs.

- Null event(impossible event) contains no outcome.

- Two events are said to be mutually exclusive if their intersection is the null event.

Ex 4.1 -------------------------------------------------------------

equally likelihood

52 (a) P(ace of spade)=1/52, (b) P(spade) = 13/52=1/4

-------------------------------------------------------------------

(relative frequency)

- lim

: A

Ex 4.2 -------------------------------------------------------------

2 . (HH, HT, TH, TT)

- P(HH)=P(HT)=P(TH)=P(TT)=

-------------------------------------------------------------------

- (Sample Space) :

- (event) : (outcome)

- (null event) :

Figure 4-1 Sample spaces. (a) Pictorial representation of an arbitrary

sample space. Points show outcomes; circles show events.

(b) Sample space representation for the tossing of two coins.

-

A or B or both :

both A and B" : (A, B)

(joint event) "A and B"

not A" :

(compound event) : ,

= disjoint sets" ( ) , event=set

- (Axiom)

Sample space S event A P(A) 0

1, P(S)=1

A, B P( )=P(A)+P(B)

(4)

4.1(a) , B C: not mutually exclusive, A B: mutually exclusive

(5)

B A Yes No

Yes

No

-

-

- P( )=P(A)+P( )=P(S)=1

- P( )= 1-P(A)

- :

=

A disjoint

=

4.1 -----------------------------------------------------------

1

2

3

4

---------------------------------------------------------------

=====================================================

- (conditional probability)

: B A

if A, B (independent) ,

(B A ),

Ex 4.3----------------------------------------------------------

P(A)=P(at least one head)=?

P(B)=P(match)=?

sol) equal likelihood -> P(A)=

, P(B)=

independent -> P(H)=

=P(T)

P(B)=P(HH)+P(TT)=

P(A|B)=P(at least one head given a match)

Bayes' rule :

so, -> A B not independent!

Ex 4.4 Independence

52

(a) P(A)=P(club), P(B)=P(black), heart, dia: red, club, spade: black

,

,

26 black 13 club

,

A, B not independent

(b) P(A)=P(king), P(B)=P(black)

A, B independent

Ex. 4.5----------------------------------------------------------

: Ch 7

?

A= (A,B) (A, ) (A,B) (A, )

P(A) =P(A,B)+

-----------------------------------------------------------

*Venn

-Venn :

: A B, A C

(6) :

ex 4.6) 52 5 . ( (face, )

(spade, heart, diamond, club)) 3 ?

) P(3 ) =

= 0.02257

4.3

(7)

(marginal probability)

- (joint event):

- : , ,

B A 1 2 i M

1

2

j

N

Ex.4.8-----------------------------------------------------------

4.2 . ?

0.1 0.4 ? ?

0.1 0.1 0.1 0.3

? 0.5 ? 1

4.1

--------------------------------------------------------------

4.2 (Random Variable )

- ( Distribution Function & Density Function).

(1) (Random Variable)

(outcome) ( ) ,

, (outcome)

.

: , 4.4(a)

: 4.4(b)

=domain, =range

(2) ( ) ( cdf: cumulative distribution function )

cdf { } .

:

lim

: right-continuous.

: non-decreasing fn.

note)

.

Note>

discrete r.v. is defined as a random variable whose cdf is right-continuous,

staircase function of x, with jumps at a countable set of points .

Probability mass function(pmf) of is a set of probabilities

of the elements in .

continuous r.v.

(3) ( pdf: probability density function )

:

note) iii) .

pdf = Probability

Ex. 4.9>

Ex. 4.10> pointer spinning experiment

r.v. : pointer

( 4.7 )

(4) (joint cdf, joint pdf)

- (joint cdf) :

- :

(marginal cdf)

(marginal pdf)

pdf

==============================

Ex 4.11) r.v. X, Y

< 4.9 >

independent

cdf

independent

conditional pdf

X, Y independent

,

==============================

Ex 4.12)

===============

joint pdf

,

marginal pdf < 4.10>

joint pdf product of the marginal pdf

Thus is not independent

===================================

(5)

- 2

Jacobian :

Ex 4.15-------------------------------------------------------

pdf :

)

Rayleigh pdf

Rayleigh pdfs are frequently used to model fading

when no line of site signal is present

4.3

(Average= ,Expectation)

discrete r.v. X value ,

-

: X

(as )

-

mean, or first moment of

, =

Ex)

(3)

-

------------------

(4)

( )

-

pdf

======================================

Ex 4.19) projectile hitting probability p

average number of projectiles fired at the target?

r.v. : projectile

r.v.

======================================

(5)

-

where

var[a]=0, a=

var[X+a]=var[X]

var[aX]=var[X]

Ex.4.20-----------------------------------------------------

---------------------------------------------------------------

(6)(7) N (linear combination of r.v.s)

r.v.s

(8) (Characteristic function)

Ex 4.21-------------------------

======================================

(9)

Z=X+Y

Ex 4.22------------------------------------

pdf of is uniform

pdf of is approaching to Gaussian (central-limit theorem)

======================================================

(10) (Covariance & Correlation Coefficient)

X Y .

)

)

4.4 (pdf)

(binomial distribution)

- ,

.

K: r.v. N A

A n k , P(K=k)?

< 4.17(a)-(d)>

(Laplace)

< 4.17(e)>

Poisson Poisson

- Poisson distribution

r.v. K: The number of occurrences(counts) of an event in a certain time period or

in a certain region in space. The Poisson r.v. K arises in the event of completely at

random.

The probability of k events in time interval T is given by (1)

Poisson - ,

T k

, (1)

Ex) r.v. K: the number of telephone call during time interval T

0 T

1 2 k nn-1

T t

When , Binomial distribution K approaches to Poisson

distribution.

i.e. n

.

< 4.17(f)>

Ex 4.24----------------------------

1000 4

P(K>3)=1-P(K

.

lim

.

: correlation coeff.

(equal pdf) equal pdf contour .

equal pdf contour =

uncorrelated Gaussian r.v.s( ) are statistically independent.

Q

:

X

* :

- :

(Chebyshev Inequality)

- 2 moment , lower bound .

- R.V X

4.49 ----------------------------------------------------

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