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