chapter 2 random variables 2.1 random variables definition. suppose that s={e} is the sampling space...
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Chapter 2 Random variablesChapter 2 Random variables
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2.1 Random variables
Definition. Suppose that S={e} is the sampling
space of random trial, if X is a real-valued fun
ction with domain S, i.e. for each eS , there
exists an unique X=X(e), then it is called that
X a Random vector.
Usually , we denote random variable by notati
on X, Y, Z or , , etc..
For notation convenience, From now on, we denote random variable by r.v.
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2.2 Discrete random variables
Definition Suppose that r.v. X assume value x1, x2, …, xn, … with probability p1, p2, …, pn, …respectively, then it is said that r.v. X is a discrete r.v. and name
P{X=xk}=pk, (k=1, 2, … )
the distribution law of X. The distirbution law of X can be represented by
X ~ P{X=xk}=pk, (k=1, 2, … ) ,
or
~XX x1 x2 … xK …
Pk p1 p2 … pk …
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(1) pk 0, k = 1, 2, … ;
(2) 1
.1k
kp=
.}{35
332
C
CCkXP
kk
==
Example 1 Suppose that there are 5 balls in a bag, 2 of them are w
hite and the others are black, now pick 3 ball from the bag without
putting back, try to determine the distribution law of r.v. X, where X
is the number of whithe ball among the 3 picked ball.
In fact, X assumes value 0 , 1 , 2 and
2. Characteristics of distribution law
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Several Important Discrete
R.V.
(0-1) distribution
let X denote the number that event A appeared in a trail,
then X has the following distribution law
X ~ P{X = k} = pk(1 - p)1 - k, (0<p<1) k = 0 , 1
or X
kp
1 0
p p1
and X is said to follow a (0-1) distribution
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Let X denote the numbers that event A appeared in a n-repeated Bernoulli experiment, then X is said to follow a binomial distribution with parameters n,p and represent it by XB ( n,p). The distribution law of X is given as :
),...,1,0(,)1(}{ nkppkXP knkk
nC
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Example A soldier try to shot a bomber with probability 0.02 that he can hit the target, suppose the he independently give the target 400 shots, try to determine the probability that he hit the target at least for twice.
Poisson theorem If Xn~B(n, p), (n = 0, 1, 2,…) and n is large e
nough, p is very small, denote =np , then
,...2,1,0,!
}{ kek
kXPk
Answer Let X represent the number that hit the target in 400 shots Then X ~ B(400, 0.02), thus
P{X2} = 1 - P{X = 0} - P {X = 1} = 1 - 0.98400-400)(0.02)(0.98399)=…
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Now, lets try to solve the aforementioned problem by putting
=np = (400)(0.02) = 8, then approximately we have
P{X2} = 1 - P{X = 0} - P {X = 1}
= 1 - (1 + 8)e - 8 = 0.996981.
Poisson distribution
X ~ P{X = k} = , k = 0, 1, 2, … (0)
e
!k
k
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Poisson theorem indicates that Poisson distribution is the
limit distribution of binomial distribution, when n is large
enough and p is very small, then we can approximate
binomial distribution by putting =np.
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Random variable Discrete r.v.
Distribution lawSeveral important r.v.s
0-1 distribution
Bionomial distributionPoisson distribution
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2.3 Distribution function of r.v.
Definition Suppose that X is a r.v., for any real number
x , Define the probability of event {Xx}, i.e. P {Xx} the distribution function of r.v. X, denote it by F(x), i.e
.
F(x) = P {Xx}.
xX
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}{lim}{0
0 xXPxXPxx
}0{ 0 xF
It is easy to find that for a, b (a<b),
P {a<Xb} = P{Xb} - P{Xa} = F(b) -F(a).
)0()(}{ xFxFxXP
For notation convenience, we usually denote distribution fucntion by d.f.
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Characteristics of d.f.
1. If x1<x2, then F(x1)F(x2);
2. for all x , 0F(x)1 , and
;1)x(Flim)(F,0)x(Flim)(Fxx
).x(F)x(Flim)0x(F 0xx
00
3. right continuous : for any x ,
Conversely, any function satisfying the above
three characteristics must be a d.f. of a r.v.
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Example1 Suppose that X has distribution
law given by the table
)(xF
x0
1
1 2
}{)( xXPxF =
X 0 1 2
P 0.1 0.6 0.3Try to determine the d.f. of X
0 0
0.1 0 1
0.7 1 2
1 2
x
x
x
x
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For discrete distributed r.v.,
X ~ P{X= xk} = pk, k = 1, 2, …
the distribution function of X is given by
xxkk
k
pxXPxF:
}{)(
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Suppose that the d.f. of r.v. X is specified as follows,
2
212
1131
1
)(
xb
xx
x
xa
xF
Try to determine a,
b and }2{},2{},10{},0{ XPXPXPXP
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Is there a more intuitive way to express the distribution
Law of a r.v. ? Try to observe the following graph
a b
?}{ bXap
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2.4 Continuous r.v.Probability density function
Definition 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
xduufxXPxF )()()( ==
then it is said that X a continuous r.v. and f(x) the density function of X , i.e. X ~ f(x) , (-<x<+)
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The geometric interpretation of density function
b
adu)u(f)bXa(P =
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2. Characteristics of density function
(1 f(x)0 , (-<x<) ;
(2) .1)( =
dxxf
(1) and (2) are the sufficient and necessary propertie
s of a density function
xaexf )(
Suppose the density function of r.v. X is
Try to determine the value of a.
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(3) If x is the continuous points of f(x), then
)()(
xfdx
xdF
Suppose that the d.f. of r.v. X is specified as follows, try to determine the density function f(x)
02
11
02
1
)(xe
xexF
x
x
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( 4 ) For any b , if X ~ f(x) ,
(-<x<) , then P{X=b} = 0 。
And
b
adxxfbXaP
bXaPbXaP
)(}{
}{}{
==
=
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Example 1. Suppose that the density function of X is specified by
Try to determine 1)the d.f. F(x), 2)P{X(0.5,1.5)}
其他0
212
10
)( xx
xx
xf
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Distribution function
Monotonicity
Standardized
Right continuous
Density function
F(x)…f(x)Nonnegative
P{a<X<b}
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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)
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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.
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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
=
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Example Suppose the age of a electronic instrument 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.
,00
05.0)(
x
xexf
0.5x
37.0)1(
1
2
0.5x edx0.5e2}P{X
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}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
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The normal distribution are one the most important
distribution in probability theory, which is widely applied
In management, statistics, finance and some other ereas.
3. Normal distribution
A
B
Suppose that the distance between A , B is ,the observed value of X is X, then what is the density function of X ?
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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 fucntion of X is specified by
2
221~ ( )
2
x
X f x e x
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(1) symmetry
the curve of density function is symmetry with respect to x= and
f() = maxf(x) = .2
1
Two important characteristics of Normal distribution
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(2) influences the distribution
, the curve tends to be flat ,
, the curve tends to be sharp ,
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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) 。
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.,2
1)( 2
2
xexx
and the d.f. is given by
xdte
xXPx
xt
,
}{)(
221
2
the density function of normal distribution is
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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.
Z~N ( 0 , 1 ) , ( 0.5)=0.6915,
P{1.32<Z<2.43}=(2.43)-(1.32)=0.9925-0.9066
注: (1) (x) = 1 - ( - x) ;
(2) 若 X ~ N(, 2) ,则
).(}{)(
x
xXPxF
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1 X~N(-1,22),P{-2.45<X<2.45}=?
2. XN(,2), P{-3<X<+3}?
EX2 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}
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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 1101 { 105} 0.42 1 0.6628 0.3371
12P X
Answer:()
120 110 100 110{100 120}
12 12
0.83 0.83 2 0.7967 1 0.5934
P X
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{ } 0.05P X x (2)令
1101 0.05
12
x
1100.95
12
x
1101.645
12
x
129.74x
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Distribution law of the function of discrete r.v.s
Distribution of the function of r.v.s
Suppose that X ~ P{X = xk} = pk, k = 1, 2, …
and y = g(x) is a real valued function, then Y = g(
X) is also a r.v., try to determine the law of Y..
Example
X
Pk
-1 0 1
31
31
31
Determine the law of Y=X2
Y
Pk
1 0
31
32
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or
Y = g(X) ~ P{Y = g(xk)} = pk , k =1, 2, …
GenerallyX
Pk
Y=g(X)
kxxx 21
kppp 21
)()()( 21 kxgxgxg
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Density function of the function of continuous r.
v.
1. If Xf(x),-< x< +, Y=g(X) , then one can try to determine the density function, one can determine the d.f. of Y firstly
FY (y) = P{Yy} = P {g(X) y} =
yxgdxxf
)()(
dy
ydFyf Y
Y
)()(
and differentiate w.r.t. y yields the density funciton
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Example Let XU(-1,1), tyr to determine the d.f. and density function of Y=X2
dxxfyF
xxgyxxf
yx
XY
X
2
2)(0
112
1
其它
ydxyFy
y
Y 2
1)(
其它0
102
1)(')(
yyyFyf YY
If y<0 0)( yFY
If 0≤y<1
If y≥1 1)( yFY
y y
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Mathematical expectation
Definition 1. If X~P{X=xk}=pk, k=1,2,…n, define
n
kkk pxXE
1
)(
the mathematical expectation of r.v. X or mean of X.
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1
||k
kk px
Definition 2. If X~P{X=xk}=pk, k=1,2,…, and
define
1
( ) k kk
E X x p
the mathematical expectation of r.v. X
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Example 2 Toss an urn and denote the points by X, try to determine the mathematical expectation of X.
6
1
1 7( )
6 2k
E X k
Definition 3 Suppose that X~f(x), -<x<, and
.)()( dxxxfXE
dxxfx )(||
the mathematical expectation of r.v. X
then define
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Example 3. Suppose that r.v. X follows Laplace distribution with density functio
n
Try to determine E(X).
x
xf exp2
1)(
dxxx
XE
exp
2)(
Let
exp | |2
xt
tt dt
dttexp0
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Mathematical expectation of several important r.v.s
1. 0-1 distribution
ppP
X
1
01EX=p
2. Binomial distribution B(n, p)
n
k
knk ppknk
nkXE
1
)1()!(!
!)(
nkppCkXP knkkn ,...1.0)1(}{
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knkn
k
ppknk
n
)1()!()!1(
!
1
)1(11
1
)1()!()!1(
)!1(
knk
n
k
ppknk
nnp
np
lnln
l
ln ppCnpkl
1
1
01 )1(1令
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3.Poisson distribution
...,2,1,0,!
}{~ kek
kXPXk
4. Uniform distribution U(a, b)
1, ,
~ ( )0, ,
a x bX f x b a
el se
0 1
1
;)!1(!
)(k k
kk
kee
kkXE
b
a
badxab
xXE ;
2)(
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5.Exponential distribution
00
0)(
x
xexf
x
1
dxexXE x
0
)(
0
xxde
dxexe xx
0
0
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6. Normal distribution N(, 2)
x,e21)x(f~X
2
2
2
)x(
dxex
XEx
2
2
2
)(
2)(
2
2 ;2
tx tt e dt
令
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EX 1 Suppose that the distribution law of X
Try to determine the mathematical expectation of Y=X2
X
Pk
-1 0 1
31
31
31
Y
Pk
1 0
31
32 3
2
3
10
3
21)( YE
Mathematical expectation of the functions of r.v.s
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Theorem 1 let X~P{X=xk}=pk, k=1,2,…,then the mat
hematical expectation of Y=g(X) is given by the follo
wing equation and denoted by E(g(X))
.)()]([)(1
kk
k pxgXgEYE
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Answer Y=ax is strictly monotonic with respect to x with inverse function
a
byyh
)(
Thus the density function of Y is given by
aa
byfyf XY
1)()(
EX2 : Suppose that r.v. X follows standardized distribution
Try to determine the mathematical expectation of Y=aX+b
dya
ey
YEa
by
1
2)( 2
2
dxe
bax x
2
2
2
b
ae
a
by
1
2
12
2
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Theorem 2 If X~f(x), -<x<, the mathematical expectation of Y=g(X) is specified as
.)()()]([)( dxxfxgXgEYE
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Suppose that X follows N(0 , 1) distribution, try
to determine E(X2), E(X3) and E(X4)
2
2
2
1)(
x
exf
dxex
XEx
22
2
2
2)(
2
2
2
x
dex
dxex
2
2
2
1
1
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dxex
XEx
23
3
2
2)(
0
dxex
XEx
24
4
2
2)(
23 2
2
x
dex
dxex x
22 2
23
3
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1. E(c)=c, where c is a constant;
2 。 E(cX)=cE(X), c is a constant.
Properties of mathematical expectation
Proof. Let X~f(x), then
dxxcxfcXE )()(
)()( XcEdxxxfc
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Example 2. Some disease will occur with probability 1% , investigate 1000 people now, it is necessary to check the blood. The method is clarified these people into ten group with each group 100 and check the mixed blood sample. If the result is negative, it is not need to do any test any more, if it is positive, then ,it is necessary to test each blood sample respectively, try to determine the average times needed for the test.
Let Xj is the number to be taken of jth group, and X the number to be taken in 1000 people, then
10,...1j
Xj
Pj
1 101
100%)99( 100%)99(1
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]99.0100
11[1000 100
)()()(10
1
10
1
j
jj
j XEXEXE
)]99.01)(101(99.0[10 100100
644
)99.01)(101(99.0 100100 jEX
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Variance
Definition 1 Suppose that X is a r.v. with EX2<∞, define E (X-EX)2 the variance of r.v. X and denote it by DX
E (X-EX)2= E [X2-2EXEX +E2X ]=EX2- E2X
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The variance of several important r.v.s
(0-1) distribution with P(X=0)=p, then DX=pq
Binomial distribution B(n,p), then DX=npq
Poisson distribution with parameter
then, DX=EX=
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The variance of several important r.v.s
Uniform distribution: Suppose that r.v. X is uniformly distributed on interval [a,b], then
Normal distribution: then
Exponential distribution: then
21( )
12DX b a
( )X E
2
1DX
2( , )X N 2DX
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Definition of mathematical expectation
Properties of mathematical expectation
Mathematical expectation of the functions of r.v.
Several important expectation