11. Finite random variables
Definition 11.1 A random
value
is called finite if the number of it’s values is finite. If these
values are:
and
,
then can consider the table
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This table
we call the series of distribution of the random variable
.
.
The set
we call the domain of the random variable
.
If we have a
series of distribution we can construct the function of distribution.
If all
are different and
,
then
Definition
11.2 Two
random variables
with the domain
and
with the domain
are independent if for any
and for any
.
Operations with random variables
Let random variables and are determined by the following series of distribution:
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Definition
11.3 If
c
is
a constant, then the random variable
is determined as follows:
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Definition 11.4 The
random variable
is called the
degree of
and is determined by the following way:
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Definition
11.5 The
random variable
is the sum of
and
.
The sum is determined by the following way.
Domain:
.
Corresponding
probabilities are:
.
Definition
11.6 The
random variable
is the difference between
and
.
The difference is determined by the following way.
Domain:
.
Corresponding probabilities are: .
Definition
11.7 The
random variable
is the product of
and
.
The product is determined by the following way.
Domain:
.
Corresponding probabilities are: .
Mathematical expectation of random variable
Let be a random variable with the series of distribution
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Definition 11.8 Mathematical expectation of is the number
.
Properties
of
If c is a constant, then
.If
are random variables a,
b are
constants, then
.
If random variables are independent, then
.
If
,
then
.If
,
then
.
.
Mathematical expectation and average of a random variable
Let
be a finite random variable with values
and values occur with the following frequencies:
occurs
times,
occurs
times,…,
occurs
times. Then
is the
average of values
.
Definition
11.9 Numbers
are relative frequencies of
.
.
So the interpretation of is the average.
Dispersion of a random variable
Definition 11.10 A dispersion of a random variable is the number
.
Remark 11.4
.
Properties of dispersion
1.
.
2. If c
is
a constant, then
.
3. If c
is
a constant, then
..
4. If c
is
a constant, then
.
5. If
and
are
independent random variables
Definition
11.11 The
number
is called the standard deviation of
.