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
11.3 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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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
.
.
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
.
12. Continuous random variables
Definition
12.1 Let
be a random variable and
be the function of distribution of
.
Then
is continuous if there exists a function p
(x)
such that
.
Definition 12.2 The function is called the integral function of distribution and p (x) is called the differential function of distribution or density of .
Properties of
1. .
2. If a domain of is (a, b), then
a)
if
;
b)
increases if
;
c)
if
.
3.
.
4.
.
Properties of p (x)
1.
.
2.
.
.
Characteristics of continuous variables
Definition
12.3 Let
be a continuous random variable with a domain
.
Then
a)
;
b)
c)
.
13. Examples of continuous random variables.
1. Unitary law of distribution
Definition
12.4 The
law of distribution of a random variable
is called the unitary law if the differential function of
distribution
is a constant for all values x
of it’s domain.
Let
be a random variable with the unitary law of distribution with
nonzero values at the interval
.
Then
for all
.
Our problem is to find C.
We
have
.
So the differential function of distribution is
.
2. Exponential law of distribution
Definition
12.5 The
law of distribution of a random variable
is called the exponential law with a parameter
if the differential function of distribution
is:
Then the integral function of distribution is
.
Applying
integration by parts we obtain
.
.
.
Theorem
12.1
.
3. Normal law of distribution
Definition
12.6 The
law of distribution of a random variable
is called the normal law with parameters
if the differential function of distribution
is:
.
Here we have the following formulas for characteristics of .
.
.
.
.
Now we study
the function
and sketch it’s graph.
a). Domain:
;
b). Range:
;
c).
;
d).
,
so we have
We obtain
that
is the point of maxima,
.
e). If we
use the substitution
,
then we obtain
.
As
is the even function then the graph of
is symmetric around the line x
= a.
Theorem 12.2 Let be the normal variable with parameters . Then
.