Download 3.2 Continuous random variable

Probability
theory
The department of math of central south
university
Probability and Statistics Course group
§3.2 Continuous random variable
1、The definitions for Continuous R.V
2、Properties for the density function of Continuous R.V
3、Properties for the distribution function of Continuous R.V
4、Common continuous R.V
1、The definitions for Continuous R.V
Definiton 3.2
A distribution function ,F(x),which describes the
probability distribution for a random variable ξ(ω)
and satisfies that there exists a non-negative Integrable
function p(x) to establish the equation
F ( x) 

x

P( y)dy
then ξ(ω)is called a continuous random
variable ，p(x) is called the probability density
function of ξ(ω), F(x) is called the distribution
function of ξ(ω).
2、Properties for the density function
of Continuous R.V
p( x)  0,  x  
(1)
(2)



p( x)dx  1
For any function, p(x),is called a probability density
function only if it satisfies the obove properties (1)、(2)
Example:
sin x

p ( x)  
 0
x  [0,  / 2]
others
For
p( x)  0
(1)
(2)




p( x)dx   sin xdx   cos x
2
0
 /2
0
Here P(x) is certainly a probability density
function
1
Geometric interpretation for distribution
finction F(x) and density function p(x)
p(x)
F (x )
F(1)
F(-1)
-1
0
1
x
3、Properties for the distribution function
of Continuous R.V
According the definition of continuous random
variable,the distribution function F(x) of R.V ξ(ω)
has the following good properties
(1)
p (a    b)  F (b)  F (a )
b
a


  p ( x)dx   p( x)dx
b
  p ( x)dx
a
(2) If F(x) is abosolute continuous,then F(x) is
continuous everywhere , has derivative in
continuous point of p(x) and
F ( x)  p( x)
(3) Especially,for any constant c,
p(  c)  F (c)  F (c  0)
 lim

c
h 0 c  h
=0
p( x)dx
Example 1
ζ is acontinuous R.Vwith the following distribution
function
x0
0

F ( x )   Ax 2 0  x  1
1
1 x

(1) What value does A gets?
(2) What is the probability that ζ gets value in
interval (0.3,0.4), that is , P(0.3< ζ <0.7)?
(3) What is the density function of X?
x0
0

F ( x )   Ax 2 0  x  1
1
1 x

Solution
(1)
Beause F(x) is left continuous at point x=1,
We get
lim F ( x)  F (1)
x 1
That is
Then
lim Ax2  F (1)  1
x 1
A=1
x0
0
 2
F ( x )   Ax 0  x  1
1
1 x

(2) P(0.3< ζ <0.7)=F(0.7)-F(0.3)
=0.7２- 0.3２=0.4
x0
0
 2
F ( x )   Ax 0  x  1
1
1 x

(3)
p( x)  F ( x)
2 x

 0
0  x 1
Others
4、Common continuous random
variables
(1). Uniform distribution
a and b are constants,and a<b，R.Vζis called
a uniform distribution random variable whose
density function is expressed as follows
 1

a xb
p( x)   b  a
 0 Others
Briefly , we write  ~ U (a, b)
For x  a,
it is obvious that F ( x)  P(  x)  0
For a  x  b
F ( x)  P (  x) 

x
a
For x  b ，obviously
1
xa
dx 
ba
ba
F ( x )  P (  x )


x
a
1
dx 
ba

b
a
x
1
dx   0dx  1
b
ba
Hence the distribution function of Uniform
distributioncan be expressed as follows
F ( x)  P (  x)
 0
x a

b  a
 1
xa
a xb
xb
The density function and distribution function
of uniform distribution
p ( x)
a
b
x
b
x
F( x)
a
(2). Normal distribution
One of the most commonly observed random
variables is one whose probability density curve is
characterized by two paramters μand σ2，the
density function is
p( x) 

1
e
2 
( x   )2
2 2
   x  
I t is called the normal random variable
marked by ζ ~ N (μ，σ2 )
Clearly, p(x) satisfies the necessary two conditions
(1)
( 2)

1
p( x) 
e
2 





p( x )  
y
( x  )2
2 2

1
e
2 
x


1
2
0
   x  



e
y2

2
( x   )2
2 2
dx
dy  1
The probability density curve is symmetrical (left half is
mirror image of the right half) and bell shaped,as shown in
the following figure.
σ
μ
Normal distribution is applicated widely .
Generally speaking, if a target is affected
by many factors the role played by each
factor is not great, then the target subjects
to normal distribution.
example :
When we Measure something , because of precision instruments,
the visual and psychological factors, outside interference and
other factors, the measurement results generally subject to
normal distribution; measurement error also subject to normal
distribution. In addition, the physical size of the adult such as
height, weight, a class in a certain area of the diameter of trees,
shells fall, certain products of a similar size, and so on are
subject to normal distribution.
On the other hand, the normal distribution has a good
nature under certain conditions, many normal distribution
can be used to approximate some other distributions, and
some can be derived through the normal distribution,
therefore, the normal distribution in the theoretical study is
also important
In fact, the normal distribution was first introduced in the
early 19th century by Gaussian (Gauss) who was studying
measurement error ,therefore normal distribution is also
known as the error distribution or Gaussian distribution;
Two parameters of p (x) ：
 — Location parameter
That is ,for given  and fifferent  , p( x)
keep same shape and lacate in different places
 — Shape parameter
For given  and different  ，p( x) locate in a
position with diffetent shapes
If 1< 2 ,
1

2  1
1
2  2
For 1, the probability that random variable
get value close to μ is larger than that of 2
Inflection point corresponding to x = μ  1
is closer to straight line x = μ than that x= μ
 2
The probabity density curve of normal distribution
1.The curve is
symmetrical to
p(x)
N (-5 , 1.2 )
0.3
0.25
0.2
line x= μ ,that
means p (μ + x)
= p(μ - x)
0.15
0.1
0.05
x= 
-8
-7
-6
-5
-4
2. p (x) gets the
maximal value
when x=μ
-3
3.Curve has infection points at points x= μ±
The relation between density curve of
normal distribution and 
When  is small
p(x)
0.3
When  become large
0.25
0.2
0.15
0.1
0.05
x= 
-8
-7
-6
-5
-4
-3
Applications
If the random variable ζ is affected by a number of
independent random factors, and the effect of each individual
is minor, and these effects can be superimposed, then ζ
subjects to normal distribution. many examples can normally
be described
Various measurement error;
People's physical characteristics;
The size of plant products;
Crop harvest;
The height of ocean wave height ;
The tensile strength of metal;
and so on
A important normal distribution：---------N (0,1) — standard normal distribution
Suppose ζ ~ N (0，1),the distributon function of ζ
should be
 ( x) 
1
2

x

e
t2

2
dt
   x  
It can easily be seen that :
( x)  1  ( x)
0.4
0.3
0.2
0.1
-x
P(| X | a)  2(a)  1
x
0.4
 (0)  0.5
0.3
0.2
0.1
-3
-2
-1
1
2
3
For a general normal distribution ：ζ ~ N (  , 2)
1 x
F ( x) 
e

2  
The distribution is :
Let
t 
s

( t  ) 2

2 2
 x  
F ( x )  

  
P(a    b)  F (b)  F (a)
P (  a )  1  F (a )
 b    a  
 
  

     
 a  
 1  

  
dt
Example2 Assume that ζ ~ N(1，4) ，
what is the probability of ζ in [0 ,1.6]
Solution
 1.6  1   0  1 
P(0    1.6)  
  

 2   2 
 0.3   0.5
 0.3  [1  0.5]
 0.6179  [1  0.6915]
 0.3094
Example 3 It is known that ~ N (2,  2 ) , and
P( 2 < ζ < 4 ) = 0.3，What is P ( ζ < 0 )?
Solution
 0  2   1   2 
 
P(  0)  

 
  
42 22
2
P(2    4)  

  0.3      (0)
     
 
2

   0.8
 
P(  0)  0.2
Graphic method
Solution
The left shaded area shown in the following figure
represents the probabity in question
0.2
0.3
0.15
0.1
0.2
0.05
-2
2
4
6
3 principle
Example 4 It is known that ζ ~ N ( α ,  2)，
what is the probability of |    | 3 ?
Solution P(|    | 3 )  P(  3      3 )
   3       3   
 
  

 
  

 3   3
 23  1 2  0.9987  1 0.9974
The αth percentile is a score that 100(1- α)% of the
distribution lies below it
Supposeζ ~ N (0，1) ，0 <  < 1，if z suffice the express
P(  z )  
Here z is called the  upper percentile．
0.4
We must remenber several important
upper percentile
0.3
0.2

0.1
z
z0.05  1.645
z0.025  1.96
(3). Exponential distribution
Suppose the random variable ζhas such a
density function with parameter λthat
e  x x  0
p( x)  
 0 x0
ζis called an exponential distribution with parameterλ,which
can be remembered as follows
 ~  ( )
 ~  ( )
We can easily get the distribution function of ζ
Firstly,When x is non-positive,it is obviously that
F ( x)  P(  x)  0
When x is greater than zero
x
F ( x)  P(  x)   e x dx
0
x
  de x  (1  e x )
0
Thus , exponential distribution function is
1  e  x x  0
F ( x)  
x0
 0
Exponential distribution which is similar to geometric
distribution has "no memory." In fact, a random variable ζ
that subjects to an exponential distribution with
parameters λ means
P(  s  t   s )  P(  s  t ,   s )
 e  ( s t )
e  s
P(  s )
 e  t
Here, s and t are arbitrary and s> 0, t> 0,
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