2.4 Continuous r.v.

Suppose that F(x) is the distribution function of r.v. X ， if there exists a no

nnegative function f(x) ， (-<x<+) ， such that for any x ， we have

( ) ( ) ( )x

F x P X x f t dt

＝ ＝

Definition2.8---P35

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

xo

)(xf

11d)(

xxfS

(1) and (2) are the sufficient and necessary properties of a de

nsity function

Note:

Properties of f(x)-----P35

;0)()1( xf;1d)()2(

xxf

, 0 3,

( ) 2 , 3 4,2

0,

kx x

xf x x

otherwi es

Suppose that the density function of X is specified by

Try to determine the value of K.

（ 3 ） For any a ， if X ～ f(x) ， (-<x<) ， then P{X=a} ＝ 0 。Proof

0 { }P X a P a x X a F a F a x

0,x then X a a x X a Therefore

Assume that

0 { } 0x F a F a x P X a

continuousF x is right

P36

}{ bXaP }{ bXaP }{ bXaP }.{ bXaP

1 2 2 14 { } ( ) ( )P x X x F x F x ;d)(2

1

xxfx

x

xo

)(xf

1 1S

1x

2x

xxfx

d)(2

Proof

.d)(2

1

xxfx

x1 2 1 2 2 1{ } { } ( ) ( )P x X x P x X x F x F x

xxfx

d)(1

}{ bXaP }{ bXaP }{ bXaP { } ( )d .b

aP a X b f x x

P35

(5) If x is the continuous points of f(x), then

)()(

xfdx

xdF

P35

. . ( ) ( )i e F x f x

Note:P36---(1)

1, 0 3,

6

( ) 2 , 3 4,2

0,

x x

xf x x

otherwi es

Example1 Suppose that the density function of X is specified by

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)∣

Suppose that the distribution function of X is specified by

0 1

( ) ln 1

1

x

F x x x e

x e

Try to determine

(1) P{X<2},P{0<X<3},P{2<X<e-0.1}.

(2)Density function f(x)

Example2

1. Uniformly distribution

if X ～ f(x) ＝ 1

,

0

a x bb a

， el se

。 。

0 a b

ab

cddxab

dxxfdXcPd

c

d

c

＝＝＝ 1

)(}{

)x(f

x

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

Several Important continuous r.v.

Example 2.14-P38

2. Exponential distribution

If X ～

0x,0

0x,e)x(f

x

＝

It is said that X follows an exponential distribution with parameter >0, the d.f. of exponential distribution is

)x(f

x

0

0,0

0,1)(

x

xexF

x

＝

Example Suppose the age of a electronic instrument 电子仪器 is X (year), which follows an exponential distribution with para

meter 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 ye

ars.

,00

05.0)(

x

xexf

0.5x

37.0)1(

1

2

0.5x edx0.5e2}P{X

}5.1|5.3{)2( XXP

37.0

1

1.5

0.5x

3.5

0.5x

e

dx0.5e

dx0.5e

}5.1{

}5.1,5.3{

XP

XXP

The normal distribution are one the most important

distribution in probability theory, which is widely applied

In management, statistics, finance and some other areas.

3. Normal distribution

A

B

Suppose that the distance between A ， B is ， the observed value of is X, then what is the density function of X ？

where is a constant and >0 ， then, X is said to follow

s a normal distribution with parameters and 2 and rep

resent it by X ～ N(, 2).

Suppose that the density function of X is specified by

2221

~ ( )2

x

X f x e x

(1) symmetry

the curve of density function is symmetry with respect to x= and

f() ＝ max f(x) ＝ .21

Two important characteristics of Normal distribution

(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 an

d 2 ＝ 1 is said to follow standard normal distributi

on and represented by X~N(0, 1) 。

.,2

1)( 2

2

xexx

and the d.f. is given by

xdte

xXPx

xt

,

}{)(

221

2

the density function of normal distribution is

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

).(}{)(

x

xXPxF

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 110Answer 1 { 105} 0.42 1 0.6628 0.3371

12P X

:（）

120 110 100 110{100 120}

12 12

0.83 0.83 2 0.7967 1 0.5934

P X

{ } 0.05P X x (2) Let

1101 0.05

12

x

1100.95

12

x

1101.645

12

x

129.74x

Example 2.15,2.16,2.17,2.18-P40-42

Homework:

P50--- Q15 ， 18

P51: 17,19,