Probability
theory
The department of math of central
south university
Probability and Statistics Course group
Chpter3 Continuous random variable
§3.1 Random variables and distribution functions
§3.2 Continuous random variable
§3.3 Multi-dimensional random variables
and distribution
§3.4 The distribution of random variable function
§3.5
The features of random variables &
Chebyshev theorem
§3.6 Conditional distribution 、expectation
and regression
§3.7
Characteristic function
§3.1
Random variables and distribution functions
1、The definition of distribution function
2、the nature of distribution function
3、The graphics of distribution function
1、The definition of distribution function
In order to describe random variables including the
discrete random variables 、 continuous random
variables and other broader types of random variables ,
a unified description method is introduced , that is , the
concept of the distribution function .
The distribution laws of discrete random variables have
been introduced before . However we want to know the
probability that a randow variable get value from a
certain interval,for example :
P{ x1≤ξ(ω)<x2 }
P{ x1≤ξ(ω) }
P{ξ(ω)<x2 }
In fact , these probabilities can be obtained through the
corresponding distribution function
Definition 3.1 A function ξ(ω)
is
defined on the
sample space Ω and get real value from a interval
of real field , (ω), is called a random variable on
the sample space Ω, saying that
F ( x)  P( ( )  x), x  (,)
is the probability distribution function, the
distribution function or distribution
For a certain r.v ξ(ω), the distribution
function is identified , a real function ,so we can
make use of the distribution function to study
random variables
For example,we may use distribution function to
calculate some probability
2、the nature of distribution function
(1) P( x1    x2 )  F ( x2 )  F ( x1 )
Here x1 and x2 are real numbers and satisfy x1 < x2
(2) P(  x )  1  F ( x)
(3) P(  x0 )  lim F ( x)  F ( x0 )
x x0 0
（4） For any real numbers x1 and x2 ,when x1 < x2
F ( x1 )  F ( x2 )
（5）
lim F ( x)  F ()  0
x  
lim F ( x)  F ()  1
x  
(6) Left continuity
For any real number x0 ,we have :
lim  F ( x)  F ( x0 )
x  x0
(7) For a discrete R,Vξ, the distribution function
satisfies：
F ( x)  P(  x)   P(  xi )
xi  x
the distribution function of a r.v ξis :
0 x  0

F ( x)  0.3 0  x  1
0.1 x  1

F(x)
P1
0
1
ξ
Fugure 3.1 the distribution function of
discrete r.v
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