Transcript (1) (x)
§4. Continuous random variables
and their density functions
Definition2.8---P35
Suppose that F(x) is the distribution function of r.v. X,if there exists a
nonnegative function f(x),(-<x<+),such that for any x,we have
F ( x)=P( X x)=
x
f (t )dt
The function f(x) is called the Probability density function (pdf)of X,
i.e. X~ f(x) , (-<x<+)
The geometric interpretation of density function
Properties of f(x)-----P35
(1) f ( x ) 0 ;
( 2)
f ( x ) d x 1;
f ( x)
Note:
S
f ( x)d x 1
1
o
x
(1) and (2) are the sufficient and necessary properties of a
density function
Suppose that the density function of X is specified by
0 x 3,
kx,
x
f ( x) 2 , 3 x 4,
2
ot her wi es
0,
Try to determine
the value of K.
P36
(3) For any a ,if X~ f(x), (<x<),then P{X=a}=0。
Proof Assume that x 0, then X a a x X a
Therefore
0 P{X a} P a x X a F a F a x
F x is right continuous
x 0 F a F a x P{X a} 0
P{a X b} P{a X b} P{a X b} P{a X b}.
P35
4
P{x1 X x2} F ( x2 ) F ( x1 )
x2
x
f ( x)d x ;
1
f ( x)
S1
1
Proof
x1 x2
o
x
P{x1 X x2 } P{x1 X x2 } F ( x2 ) F ( x1 )
x2
x1
x2
x1
f ( x) d x f ( x) d x
f ( x ) d x.
b
P{a X b} P{a X b} P{a X b} P{a X b} a f ( x)d x.
P35
(5) If x is the continuous points of f(x), then
dF ( x )
f ( x) i.e.F ( x) f ( x)
dx
Note:P36---(1)
Example1
Suppose that the density function of X is specified by
1
0 x 3,
6 x,
x
f ( x) 2 , 3 x 4,
2
ot her wi es
0,
Try to determine 1)the value of K
2)the d.f. F(x),
3)P(1<X≤3.5)
4)P(x=3)
5)P(x>3.5∣x>3)
Example2
Suppose that the distribution
function of X is specified by
x 1
0
F ( x) ln x 1 x e
1
xe
Try to determine
(1) P{X<2},P{0<X<3},P{2<X<e-0.1}.
(2)Density function f(x)
Several Important continuous r.v.
f (x)
1. Uniformly distribution
1
if X~f(x)= b a , a x b
0, el se
0
。
。
a
b
It is said that X are uniformly distributed in
interval (a, b) and denote it by X~U(a, b)
For any c, d (a<c<d<b),we have
1
d c
P{c X d }= f ( x)dx=
dx=
c
c ba
ba
d
d
x
Example 2.14-P38
2. Exponential distribution
f (x)
e x , x 0
If X~ f ( x )=
0, x 0
x
0
It is said that X follows an exponential
distribution with parameter >0, the d.f. of
exponential distribution is
1 e x , x 0
F ( x)=
0, x 0
Example Suppose the age of a electronic instrument电子仪器
is X (year), which follows an exponential distribution with
parameter 0.5, try to determine
(1)The probability that the age of the instrument is more than 2
years.
(2)If the instrument has already been used for 1 year and a half,
then try to determine the probability that it can be use 2 more
years.
0.5e0.5x
f ( x)
0
x0
x 0,
(1)P{X 2} 0.5e0.5xdx e 1 0.37
2
( 2) P{ X 3.5 | X 1.5}
P { X 3. 5, X 1 . 5 }
P { X 1.5}
0.5x
0.5e
dx
3.5
0.5x
0.5e
dx
1.5
e
1
0.37
3. Normal distribution
The normal distribution are one the most important
distribution in probability theory, which is widely applied
In management, statistics, finance and some other areas.
B
A
Suppose that the distance between A,B is ,the
observed value of is X, then what is the density
function of X ?
Suppose that the density function of X is specified by
1
X ~ f ( x)
e
2
x
2 2
2
x
where is a constant and >0 ,then, X is said to follows
a normal distribution with parameters and 2 and
represent it by X~N(, 2).
Two important characteristics of Normal distribution
(1) symmetry
the curve of density function is symmetry
with respect to x= and
f()=max f(x)=
1
.
2
(2) influences the distribution
,the curve tends to be flat,
,the curve tends to be sharp,
4.Standard normal distribution
A normal distribution with parameters =0 and
2=1 is said to follow standard normal distribution
and represented by X~N(0, 1)。
the density function of normal distribution is
1
e
2
( x)
x2
2
, x .
and the d.f. is given by
( x ) P { X x }
1
2
x
e
t2
2
dt , x
The value of (x) usually is not so easy to compute
directly, so how to use the normal distribution table
is important. The following two rules are essential
for attaining this purpose.
Note:(1) (x)=1-(-x);
(2) If X~N(, 2),then
F ( x ) P{ X x } (
x
).
1 X~N(-1,22), P{-2.45<X<2.45}=?
2. XN(,2), P{-3<X<+3}?
EX 2 tells us the important 3 rules, which are widely
applied in real world. Sometimes we take
P{|X- |≤3} ≈1 and ignore the probability of
{|X- |>3}
Example The blood pressure of women at age 18 are
normally distributed with N(110,122).Now, choose a
women from the population, then try to determine (1)
P{X<105},P{100<X<120};(2)find the minimal x such that
P{X>x}<0.05
105 110
Answer: ()
1 P{ X 105}
0.42 1 0.6628 0.3371
12
120 110
100 110
P{100 X 120}
12
12
0.83 0.83 2 0.7967 1 0.5934
( 2) Let P{X x} 0.05
x 110
1
0.05
12
x 110
0.95
12
x 110
1.645
12
x 129.74
Example 2.15,2.16,2.17,2.18-P40-42
Homework:
P50--- Q15,18
P51: 17,19,