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Title: Probability and Statistics
Description: Subject Contents are for 2nd and 3rd year students. Chapter:-2 Random Variables.
Description: Subject Contents are for 2nd and 3rd year students. Chapter:-2 Random Variables.
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Chapter 2
Random Variables
Lectures 4 - 7
In many situations, one is interested in only some aspects of the random experiment
...
The probability space corresponding to the experiment of tossing
coins is given by
and
is described by
Our interest is in noting
, i
...
, in the map
...
But all functions defined on the sample space
are not useful, in the sense that we may not be able to assign probabilities to all basic events associated
with the function
...
This
motivates us to define random variables
...
1 Let
be a probability space
...
0
...
...
Example 2
...
15
Let
For
Since
is a random variable
...
Define
...
0
...
0
...
is a random variable imply that the basic events
this imply a larger class of events associated with
examine this, one need the following
-field
...
2 The
of subsets of
associated
can be assigned probabilities (i
...
in
-field generated by the collection of all open sets in
and is denoted by
are in
is called the Borel
...
Lemma 2
...
2
Let
...
For
,
Therefore
Hence
For
,
Therefore
Also, for
Therefore,
contains all open intervals
...
Thus,
This completes the proof
...
0
...
Then
is a random variable on
iff
for all
Proof
...
is a random variable
...
e
...
0
...
0
...
0
...
Hence
Note that
...
Now from (2
...
2) and Lemma 2
...
1, it follows that
for all
...
e,
...
Remark 2
...
1
Theorem 2
...
3 tells that if
for all
Theorem 2
...
4
Then
Proof
...
be a random variable
...
for all
, then
Hence
...
Similarly from
it follows that
This completes the proof
...
3 The sigma field
Remark 2
...
2
is in
is called the sigma field generated by
The occurrence or nonoccurence of the event
or not
...
tells whether any realization
collects all such information
...
Example 2
...
16
Theorem
2
...
5
Let
...
Proof: The proof of first two are simple exercises
...
Therefore
is a random variable
...
Note that
Since
are random variables and
is their sum,
variable
...
0
...
is a sequence of random variables and
is a random variable
...
(i) Set
Therefore
...
The proof of
is a random variable is similar
...
0
...
Proof
...
such that
is a random
Hence
Now suppose
If
, then there exists there exists
and
such that
(2
...
3)
(This follows, since
)
Also
This contradicts (2
...
3)
...
0
...
is a
-field, using(2
...
4) it follows from the definition of
-field
Title: Probability and Statistics
Description: Subject Contents are for 2nd and 3rd year students. Chapter:-2 Random Variables.
Description: Subject Contents are for 2nd and 3rd year students. Chapter:-2 Random Variables.