Information Technology Reference
InDepth Information
•
F
∈
F
\
∈
F
The event space
is closed under complement, i.e., if
E
, then
Ω
E
.
Given an event space
F
, the probability measure
P
must satisfy certain axioms.
•
For all
E
∈
F
,
P(E)
≥
0.
•
P(Ω)
=
1.
•
P(E
2
)
.
Random variables play an important role in probability theory. Random variables
allow us to abstract away from the formal notion of event space, because we can
define random variables that capture the appropriate events that we are interested in.
More specifically, we can regard the random variable as a function, which maps the
event in the outcome space to real values. For example, suppose the event is “it is
sunny today”. We can use a random variable
X
to map this event to value 1. And
there is a probability associated with this mapping. That is, we can use
P(X
For all
E
1
,E
2
∈
F
,if
E
1
∩
E
2
=∅
,
P(E
1
∪
E
2
)
=
P(E
1
)
+
=
1
)
to
represent the probability that it is really sunny today.
21.1.2 Probability Distributions
Since random variables can take different values (or more specifically map the event
to different real values) with different probabilities, we can use a probability distri
bution to describe it. Usually we also refer to it as the probability distribution of the
random variable for simplicity.
21.1.2.1 Discrete Distribution
First, we take the discrete case as an example to introduce some key concepts related
to probability distribution.
In the discrete case, the probability distribution specifies the probability for a
random variable to take any possible values. It is clear that
a
P(X
=
=
1.
If we have multiple random variables, we will have the concept of joint distribu
tion, marginal distribution, and conditional distribution. Joint distribution is some
thing like
P(X
a)
=
=
b)
. Marginal distribution is the probability distribution of
a random variable on its own. Its relationship with joint distribution is
a,Y
P(X
=
a)
=
P(X
=
a,Y
=
b).
b
Conditional distribution specifies the distribution of a random variable when the
value of another random variable is known (or given). Formally conditional proba
bility of
X
=
a
given
Y
=
b
is defined as
P(X
=
a,Y
=
b)
P(X
=
a

Y
=
b)
=
.
P(Y
=
b)