probability theory school of mathematical science and computing technology in csu course groups of...
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Probability Theory
School of Mathematical Science and
Computing Technology in CSU
Course groups of Probability and Statistics
§2.4 Figure characteristics of random variables
Distribution list is able to describe the statistical characteristics
of random variables completely , However, in some practical p
roblems , only need to know some characteristics of random v
ariables and thus do not need to derive a result of its distribution
function .For example :
Assessment of the viability of an enterprise, only need to
know the level of per capita profit of the enterprises ;
Study the merits of rice varieties, we are concerned abou
t the average rice grains and the average weight of each piec
e costs ;
Test the quality of cotton, they should not only pay attenti
on to the average length of fiber, but also pay attention to th
e deviate degree between the length of fiber and the average
length, the longer the average length and the smaller the dev
iate degree, the better the quality.
Study the level of one shooter, we not only depend on his av
erage ring number whether high or not, but also depend on hi
s scope of impacts whether small or not, that is, whether the f
luctuations in data small.
From the above example we can see , some values relating to random variables. Although we ca
n not completely describe the random variable , but we
can clearly describe the important feature of random vari
ables in some respects. Characteristics of these figures h
ave great significance both in theory and practice.
One aspect of probability characteristics of
random variable are available to describe
by figures.
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The average values of random varia
bles ——Mathematical expectation The situation of random variable val
ues are deviate from the mean value on a
verage—— Square
The contentof this section
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Definition: Suppose discrete distribution of random variable X as
,2,1,)( kpxXP kk
If the infinite series
1kkk px
absolute convergence ,then called which the sum is random variable X as mathematical expectation ,recorded as
1k
kk pxEX
1.The definition of mathematical expectation Section I Mathematical expectation
n
k
knkkn ppkCEX
0
)1(
n
k
knk ppknk
nk
1
)1()!(!
!
n
k
knk ppknk
nnp
1
)1()1(1 )1()!()!1(
)!1(
1
0
)1(1 )1(
n
k
knkkn ppCnp
np
EXpnBX Calculated),,(~Known
Answer
Example 1
EXPX Calculated),0(),(~Known
Example 2
Answer
EX
e
kk
k
k
0 !
1
1
)!1(k
k
ke
Common mathematical expectation of random variable Distribution Expectation
pXP
pXP
1)0(
)1(p
B(n,p)nk
ppCkXP knkkn
,,2,1,0
)1()(
np
P()
,2,1,0!
)(
kk
ekXP
k
Probability distribution
Parameters for the 0-1 distribution of p
2. The nature of mathematical expectation
following)
in then expectatio almathematicexist meet we
variablerandom theconstant
is ,, variablerandom is, uppose
suppose We( S cbaYX ,
cbEYaEXcbYaXE )()1(
0then ,0 uppose)2( EXXS
bEXabXa hen , uppose)3( tS
n
i
n
iii
i
EXXE
niX
EXEYXYE
YX
1 1
is then thereothereach of
t independen are21If:Promotion
)(Then variablesrandom
t independenmutually twoare Suppose)4(
)(,
),,(,
,
Prove : Only prove Nature (4) at n=2
,2,1
ason distributi its record And
and of valuespossible all as recorded
and usely respectivevariable
random discrete a is that assumeWe
21
21
2121
jipbXaXP
XX
bbaa
X
ijji
i
,),(
,,,,,,
may
:known weassumptiont independen theFrom
)()( 21 jiij bXPaXPp
21
21
,21
,21
21
21
)()(
)()()(
,
EXEX
bXPbaXPa
bXPaXPbapbaXXE
baXX
bXaX
jjj
iii
jijiji
jiijji
ji
ji
so
, is there
, whenBecause
Answer Import the random variables
Example 7 A Civil Aviation bus contains 20 visitors leave th
e airport , the visitors can get off at 10 stations , if one sta
tion has no passengers to get off the bus will not stop , tak
e X as the number of stops , Calculate EX ( Suppose each
passenger get off at various stations have the same possibilit
y, and suppose whether the passengers get off or not are ind
ependent of each other )
istation at offget someone1
istation at offget nobody ,0
,iX
10,,2,1 i
Then there is
1021 XXXX
10
9 is istation at offget not didpassenger
oneany ofy probabilit the titleby the Intended ,
20)10
9( is istation at off
getnot do passengers 20 ofy probabilit theSo
20)10
9(1 is i station
at offget someone of yprobabilit The
2020 )10
9(1)1(,)
10
9()0( ii XPXP
20)10
9(1)1(1)0(0 iii XPXPEX
So
10
11021 )(
iiEXXXXEEX
784.8])10
9(1[10 20 ( time
s)
That is
time
Arrived
yProbabilit
50:930:910:9
50:830:810:8
6
2
6
3
6
1
Example 8 According to regulation , one st
ation everyday 8 : 00 ~ 9:00,9:00 ~ 10:00 bot
h happen to have a bus reach the station , bu
t the reach time is random , and the arrive ti
me are independent of each other , the law is
. time waitinghis of
n expectatio almathematic thecalculate
00,:8at station reach theA visitor )1(
Answer
minute)
by counted( is visitor the
of time waitingthe Suppose
X
. time waitinghis of
n expectatio almathematic thecalculate
20,:8at station reach theA visitor )2(
is X ofon distributi The)1(
X
kp
503010
6
2
6
3
6
1
)points(33.33 EX
X
kp
9070503010
6
1
6
2
6
1
6
3
6
1
6
1
6
2
6
3
is X ofon distributi The)2(
example for,above tableOn the
6
3
6
1)()()()70( BPAPABPXP
,: "108at arrive busfirst The" case theisA
thoseAmong
,: "309at rrive bus second he" case theis aT B
)min(27.2236
290
36
370
36
150
6
230
6
310
EX
then,)(
if variablesrandom discreteFor
kk pxXP ,X
3. The mathematical expectation of random variable function
exists)]([)(
function continuousits variables
random a is thereSupposeTheorem
xgEXgY ,X,
)]([ XgE k
kk pxg )(
)function continuous is(,),(.,
variablerandom offunction theisSuppose
gYXgZYX
Z
is lawon distributi theY) (X,
variablerandom discrete ldimensiona- two theIf
,2,1,,),( jipyYxXP ijji
ijji pyxgYXgEEZ ),()],([
isthereThen
X 1 3
P 3/4 1/4
Y 0 1 2 3
P 1/8 3/8 3/8 1/8
X
1 0 3/8 3/8 0
3 1/8 0 0 1/8
Y 0 1 2 3
)(,,Calculate XYEEYEX
Answer
Example 9 Known the joint distribution of( X,Y ) is
8
1)33(0)23(0)13(
8
1)03(
0)31(8
3)21(
8
3)11(0)01()(
XYE
4
9
2
3,
2
3 EYEX
For a particular disease survey , n indi
viduals need a blood test , blood tests
can be two ways:
(1)Tests separately for each person's bl
ood , need to test n times totally ;
Blood program selection
33 、、 Simple Simple applicationapplication of of mathematica mathematica
l expectationl expectation
Suppose the probability of tested positive is p i
n someone area , and each is a positive person ar
e independent of each other. Try to select a method
which can reduce the number of tests. .
K individuals will be mixed with the blood te
sts , if the test results become negative , then the
k individual blood tests only once ; If the results
become positive, then the k individuals will have b
lood test one by one to identify sick persons, then
k individual blood tests to be k + 1 times.
Answer For the simple calculation , Based n are multiples of k , suppose divided into a total of n / k group, the number of tests for group i
required to be Xi
kp1 kp 11
Xi
P
1 k + 1
]11)[1(1 kki pkpEX
kpkk 1)1(
k
n
iiEXEX
1
kpkkk
n 1)1(
kpn k 1
)1(1
,01
)1(
kp kIf Then EX < n
Such as ,
10000110010
1999.0110000
,10,001.0,10000
10
EX
kpn
Section II SquareGuide example Test the quality of two groups of light bulbs, which were randomly selected 5, the measured lifetime (unit: hours) as follows:
A: 2000 1500 1000 500 1000
B: 1500 1500 1000 1000 1000
Let us compare the quality of these two groups of light bulbs
After calculated :Average life are :A:1200 B:1200
After Observated :A has large departurein useful life,B has small departurein useful life,so,B has better quality
Mathematical expectation Square
1. The definition of square
is that,,
called then,
,
DXX
EX)E(XEX)E(X
X22
as recordedof square theas
exist
if variablerandom a is Suppose:Definition
(X - EX)2 —— Random variable X the value
of deviation from the average of the situation
are a function of X is also a random variable E(X - EX)2 —— Random variable X the value
of the average deviation from the average
deviation from the degree - a number Note:
Variance reflects the random variable relative degree of its deviation from the mean.
2)(Square EXXEDX DXdeviation) (standard squareMean
,2,1,)( kpxXP kk
X is discrete random variables, probability distribution is:
k
kk pEXxDX 2
If X is continuous random variables, probability density is f (x)
dxxfEXxDX )(2
Commonly used formula for calculating the variance:
22 )(EXEXDX
2. The nature of the square )Constant is(0)1( CDC
DXCCXD 2)()2(
DYDXYXD
YX
)( variablesrandom
t independenmutually twoare,Suppose)3(
then,
1)(isThat ,)Here(constant
one takingofy probabilit thetois0
toconditions sufficient andnecessary The)4(
CXPEXCC
XDX
)constantsarbitrary is(
)()()5( 22
a
aEXaXEDX
Example1 Suppose X ~ P (), Calculate DX.
Answer
0 !k
k
k
ekEX
1
1
)!1(k
k
ke
3. Square calculation
EXXXEEX )]1([2
!)1()]1([
0 k
ekkXXE
k
k
2
2
22
)!2(
k
k
ke
22EX
22 )(EXEXDX
Example 2 Suppose X ~ B( n , p) , calculate DX Answer One follow the example above
calculate DX
Answer Two Import the random variables nXXX ,,, 21
happen matter times i of est
happenA matter times i of est
AT
TX i ,0
,1
nXXX ,,, 21 are independent of
ni ,,2,1 )1( ppDX i
n
iiXX
1
so,
)1(1
pnpDXDXn
ii
each other ,
Common random variable of variance
Distribution SquareProbability distribution
Parameters for the 0-1 distribution of p pXP
pXP
1)0(
)1(p(1-p)
B(n,p)nk
ppCkXP knkkn
,,2,1,0
)1()(
np(1-p)
P()
,2,1,0!
)(
kk
ekXP
k
Example 8 Suppose X express the requir
ed fire number of shooting independent
until hits the target n times. Known for e
ach target shooting in a probability of p ,calculate EX , DX
Answer X i express the required fire numbe
r of hit the target i - 1 times to hit the target
i times , i = 1,2,…, n
1,2,1,)( 1 qpkpqkXP ki
1
1
1
1
k
k
k
ki kqpkpqEX
pqp
1
)1(
12
nXXX ,,, 21 are independent of
n
iiXX
1
each other , moreover
1
1
1
12 )1(k
k
k
ki kpqpqkkEX
pqkkpq
k
k 1)1(
2
2
px
dx
dpq
qxk
k 1
02
2
pxpq
qx
1
)1(
23
2
2
p
p
p
nEXEX
n
ii
1
Therefore,
21
)1(
p
pnDXDX
n
ii
Only know the expectations of random variables
and the variance can not determine their
distribution, such as:
222
112
p
p
pp
pDX i
X
P
-1 0 1
0.1 0.8 0.1
Y
P
-2 0 2
0.025 0.95 0.025
and
2.0,0 DXEX
2.0,0 DYEY
They have the same expectation and variance, but the distribution is different .