ref: F. G. Stremler, Introduction to Communication Systems 3/e
• Probability
• All possible outcomes (A
1
N 
P ( A i
)

1 i

1
• Joint probability
P ( A )
 lim
N
 
P ( AB
P ( AB )

P ( B |
N
A
N to A
N
) are included
)
 lim
N
 
A ) P ( A )

N
N
P (
AB
A | B ) P ( B )
• Conditional probability
P ( B | A )

N
AB
N
A

N
AB
N
N
A
/ N

P ( AB )
P ( A )
P ( A | B )

N
AB
N
B

N
AB
N
N
B
/ N

P ( AB )
P ( B )
Ya Bao Fundamentals of Communications theory 1

• Bayes’ theorem
P ( B | A )

P ( B ) P ( A | B )
P ( A )
• Random 2/52 playing cards. After looking at the first card, P(2 nd is heart)=? if 1 st is or isn’t heart
• Probability of two mutually exclusive events
P(A+B)=P(A)+P(B)
• If the events are not mutually exclusive
P(A+B)=P(A)+P(B)-P(AB)
Ya Bao Fundamentals of Communications theory 2

• A real valued random variable is a real-value function defined on the events of the probability system.
• Cumulative distribution function (CDF) of x is
F ( a )

P (
• Properties of F(a) x
 a )
 n lim
 
( n x n
 a
)
• Nondecreasing,
• 0<=F(a)<=1,
F (

)

0
F (

)

1
Ya Bao Fundamentals of Communications theory 3

f ( x )
 dF ( a )
| a
 x da
Properties of PDF f ( x )

0 .
 
  f ( x ) dx

F (

)

1
Ya Bao Fundamentals of Communications theory 4

• Discrete: random variable has M discrete values
CDF or F(a) was discontinuous as a increase
Digital communications
PDF f ( x )
 i
M 

1
P ( x i
)

( x
 x i
)
M is the number of discretely events
CDF F ( a )
 i
L 

1
P ( x i
)
L is the largest integer such that x
L
 a , L

M
Ya Bao Fundamentals of Communications theory 5

• Continuous distributions: if a random variable is allowed to take on any value in some interval.
CDF and PDF would be continuous functions.
Analogue communications, noise.
• Expected value of a discretely distributed random variable y

[ h ( x )]
 i
M 

1 h ( x i
) P ( x i
)
Normalized average power
P =
 i
y i
2
p(y i
)
Ya Bao Fundamentals of Communications theory 6

• Binomial
• Poisson
• Uniform
• Gaussian
• Sinusoidal
Ya Bao Fundamentals of Communications theory 7

Ya Bao Fundamentals of Communications theory 8