1.
p
(
x
)
≥
0 for all
x
, and
2.
∑
x
p
(
x
) = 1
.
For example, if
X
is the random variable representing the outcome
of rolling a fair die, the random variable
X
has a set of possible values
{
1
,
2
,
3
,
4
,
5
,
6
}
each with probability
p
(
x
) = 1
/
6.
If
X
drawn is from dis
tribution F, we write
x
∼
F
. In this die example,
X
is a discrete random
variable because it takes one of a set of discrete values. Random variables
can also be continuous; they can take on any of a set of continuous val
ues. Here we will treat discrete random variables first, and then move on to
continuous random variables.
1.2
Events
Closely related to the notion of a random variable is the concept of an event.
Definition 2
An
event
is the case that a random variable takes on a value
within a described subset of possible values.
For example, the event that I roll a die and get an odd number can be
represented as the event
X
∈ {
1
,
3
,
5
}
. Events can include single values of
random variables, e.g. the event that
X
= 4; they can include all possible
values
X
∈ {
1
,
2
, . . . ,
6
}
, and they can include no possible values
X
∈ {}
.
If events
A
1
, A
2
, . . . , A
n
are mutually exclusive events with probabilities
P
[
A
i
], the probability that any one of them occurs is
P
[
A
1
or
A
2
. . .
or
A
n
] =
X
i
P
[
A
i
]
.
1.3
Conditional probability
Conditional probability lets us talk about the chance that one event occurs
given that another occurs.
Definition 3
The
conditional probability
of A given B is
P
[
A

B
] =
P
[
A
and
B
]
P
[
B
]
2