1
Lecture 3
Random Variables and their
Statistics
Random Variables
Recall we wanted an algebraic structure to
combine various possible events from the
sample space
Ω
σ−
field .
To use the computational engine of calculus
and set up a deductive ( conclude and
predict) framework
We map events from sample space to
\
Definition of sets on
How should sets and subsets be represented
To accommodate unions and intersections?
A function of one argument can be defined?
Representation of subsets of
by
:
Ex1:
Suppose we want to represent a set
with
sets.
\
\
(
, ]
x
−∞
(
,
]
a b
(
,
]
x
−∞
Continued
Ex2:
Ex3:
(
,
]
a b
(
, ]
(
,
]
b
a
= −∞
− −∞
(
,
]
(
,
]
c
b
a
= −∞
−∞
∩
1
1
{ }
lim{(
,
)}
n
a
a
a
n
n
→∞
=
−
+
[
,
]
(
,
]
{
}
a
b
a b
a
=
∪
1
1
{ (
,
]
(
,
]
{lim (
,
]}
c
n
b
a
a
a
n
n
→ ∞
=
− ∞
− ∞
−
+
∩
∪

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*