Axioms of Probability
•
A probability measure
P
satisfies the following axioms:
1.
P(
A
)
≥
0
for every event
A
in
F
2.
P(
Ω
) = 1
3.
If
A
1
, A
2
, . . .
are
disjoint events
— i.e.,
A
i
∩
A
j
=
∅
, for all
i
=
j
— then
P
∞
i
=1
A
i
=
∞
i
=1
P(
A
i
)
•
Notes:
◦
P
is a
measure
in the same sense as
mass
,
length
,
area
, and
volume
— all
satisfy axioms 1 and 3
◦
Unlike these other measures,
P
is bounded by 1 (axiom 2)
◦
This analogy provides some intuition but is not su
ffi
cient to fully understand
probability theory — other aspects such as conditioning and independence are
unique to probability
EE 278: Basic Probability
1 – 3
Discrete Probability Spaces
•
A sample space
Ω
is said to be
discrete
if it is countable
•
Examples:
◦
Flipping a coin:
Ω
=
{
H, T
}
◦
Rolling a die:
Ω
=
{
1
,
2
,
3
,
4
,
5
,
6
}
◦
Flipping a coin
n
times:
Ω
=
{
H, T
}
n
, sequences of heads/tails of length
n
◦
Flipping a coin until the first heads occurs:
Ω
=
{
H, TH, TTH, TTTH, . . .
}
◦
Number of packets arriving at a node in a communication network in time
interval
(0
, T
]
:
Ω
=
{
0
,
1
,
2
,
3
, . . .
}
The first three examples have
finite
Ω
, whereas the last two have
countably
infinite
Ω
. Both types are considered
discrete
•
For discrete sample spaces, the set of events
F
can be taken to be the set of all
subsets of
Ω
, sometimes called the
power set
of
Ω
EE 278: Basic Probability
1 – 4