Probability & Independence

• Sample space

• Random variable

• Probability

face Y f

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

Random variable

Sample point ( ): the object where selection is made

Sample space ( ) : the set of all sample points

},...,,,,{ 3621000

Random variable ( , , …. ):X

• a characteristic (number, color, etc.) of sample points.• a function converts sample points to their characteristics.• a function from sample space to the set of real numbers.

Y

RY :

"")("")( 10 redXgreenX

"1")("00")( 100 YY

1.....,, 32

)(Y

)( 1Y

Random variables

No. Color Even … f

00 Green - 1/38

0 Green - 1/38

1 Red X 1/38

2 Black O 1/38

3 Red X 1/38

4 Black O 1/38

1/38

35 Black X 1/38

36 Red O 1/38

123

4

35

36

000

• Random variable generates subsets of the sample space.

""greenX },{ 0000

33Y },,,{ 363534331

oddisY },,,{ 35312

• Event: a subset of the sample space

AB

CE

• Each point in layout corresponds to an event.

}{..}8{ 8eiA

},,,{..}14,13,11,10{ 14131110 eiB

]1,0[: SP}8{A

SBA ....,,

)(P

38

1)( AP

• Probability is a function converts events to a number between 0 and 1.

eventsallofsetS :

}14,13,11,10{B

38

4)( BP

1)( P

0)( P

cA A: complementary event of

)(1)( APAP c

: sample space, the total set, the universal event

c : null event

)()()()( BAPBPAPBAP

A: the set of even numbers

B : multiples of 3

BA : multiples of 6

BA : multiples of 2 or 3

1214

15

16

18

10 20

21

22

24

26

27

28

3032

33

3436

2

3

4

6

8

9

0

00

11

1317

19

2325 29

31

35

157

A

B

38

6

38

12

38

18

38

24)( BAP

38

12)( BP

38

18)( AP

38

6)( BAP

38

12)( BP

18

4)|( DBP

38

18)( AP

38

6)( BAP

18

10)|( DAP

38

2)|( DBAP

15

24

33

14 16 32 34 12 18 30 36

610 20 22 26 282 4 8

21 273 9

11 13 17 29 31 35

19 23 251 5 7

B

A

D

0 00

RY :

S

]1,0[: SP

: sample space, the total set, universal event,

roulette, chocolate box, urn

: set of subsets of

layout

Roulette Holes Characteristics of holes

sample points, sample space random variables

Layout

set of subsets of sample space

Y

S

X f

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

(X,Y)

f

(1,1) 1/36

(1,2) 1/36

(1,3) 1/36

… 1/36

(6,5) 1/36

(6,6) 1/36

}6,5,4,3,2,1{ )}6,6(....,),2,1(),1,1{(

How to get money from casino ?

Bet $100 + “all the amount you lose” every time.

Think coin tossing game.

Then you will win at least a time, then stop there.

$500

House margin Using chips Betting limit

“Who are willing to play a gambling game, seeing rising sun?”

How to get money from casino ?

The best strategy is the one that casinos want to keep out of.

That is the strategy leaving casino as soon as possible.

Counterplots of casino:

High quality accommodation

Far away location

• Independence

• Combinatorics

• Joint distribution

• Conditional distribution

(X,Y)

f

(1,1) 1/36

(1,2) 1/36

(1,3) 1/36

… 1/36

(6,5) 1/36

(6,6) 1/36

X\Y 1 2 … 6 Tot

1 1/36

1/36

1/36 1/6

2 1/36

1/36

1/36 1/6

… … …

6 1/36

1/36

1/36 1/6

Tot 1/6 1/6 1/6 1.0

Joint dist’n, Marginal dist’n

)(xf ),( yxf

X\Y 1 2 3 4 5 6 Tot

1(H) 2/18

2/18 2/18 2/18 2/18 2/18 2/3

0(T) 1/18

1/18 1/18 1/18 1/18 1/18 1/3

Tot 1/6 1/6 1/6 1/6 1/6 1/6 1.0

),( yxf

X\Y 1 2 3 4 5 6 Tot

1(H)

0(T)

Tot 1.0

)(xf

)(yf

Independence of random variables

X\Y 1 2 3 4 5 6 Tot

1(H) 2/18

2/18 2/18 2/18 2/18 2/18 2/3

0(T) 1/18

1/18 1/18 1/18 1/18 1/18 1/3

Tot 1/6 1/6 1/6 1/6 1/6 1/6 1.0

)()(),( yfxfyxf

X\Y 1 2 3 4 5 6 Tot

1(H) 2/18

1/18 2/18 1/18 2/18 1/18 1/2

0(T) 1/18

2/18 1/18 2/18 1/18 2/18 1/2

Tot 1/6 1/6 1/6 1/6 1/6 1/6 1.0

)()(),( yfxfyxf

YX

YX

joint pdf, marginal pdf, independence

X\Y 0 1 2 T

0 1/16

1/81/16

1/4

1 1/8 1/4 1/8 1/2

2 1/16

1/81/16

1/4

T 1/4 1/2 1/4 1.0

X\Y 0 1 2 T

0 1/8 1/8 0 1/4

1 1/8 1/4 1/8 1/2

2 0 1/8 1/8 1/4

T 1/4 1/2 1/4 1.0

X\Y 0 1 2 T

0 0 1/4 0 1/4

1 1/4 0 1/4 1/2

2 0 1/4 0 1/4

T 1/4 1/2 1/4 1.0

1),(0 ji yxf

1),( i j

ji yxf

0),( yxf

1),( dydxyxf

j

jii yxfxf ),()(

i

jij yxfyf ),()(

dyyxfxf ),()(

dxyxfyf ),()(

)|()()|()(),( xyfxfyxfyfyxf

)(

),()|(

yf

yxfyxf

conditional distribution f(x|y)

X\Y 0 1 2 T

0 1/16

1/81/16

1/4

1 1/8 1/4 1/8 1/2

2 1/16

1/81/16

1/4

T 1/4 1/2 1/4 1.0

X\Y 0 1 2 T

0 1/8 1/8 0 1/4

1 1/8 1/4 1/8 1/2

2 0 1/8 1/8 1/4

T 1/4 1/2 1/4 1.0

X\Y 0 1 2

0 1/4 1/4 1/4

1 1/2 1/2 1/2

2 1/4 1/4 1/4

T 1.0 1.0 1.0

X\Y 0 1 2

0 1/2 1/4 0

1 1/2 1/2 1/2

2 0 1/4 1/2

T 1.0 1.0 1.0

YX )()(),( yfxfyxf

X\Y 0 1 2 T

0 1/16

1/81/16

1/4

1 1/8 1/4 1/8 1/2

2 1/16

1/81/16

1/4

T 1/4 1/2 1/4 1.0

X\Y 0 1 2 T

0 1/4 1/2 1/4 1.0

1 1/4 1/2 1/4 1.0

2 1/4 1/2 1/4 1.0

X\Y 0 1 2

0 1/4 1/4 1/4

1 1/2 1/2 1/2

2 1/4 1/4 1/4

T 1.0 1.0 1.0

),( yxf )(xf

)(yf

)|( yxf )|( xyf

)])([(),( YX YXEYXCov

)()()( YEXEXYE

YXYX XYXYE

YXYX XEYEXYE )()()(

YXXYE )(

0)()()(),( YEXEXYEYXCov

YX )()(),( yfxfyxf

j

jijii

yxfyxXYE ),()(

j

jijii

yfxfyx )()(

)()()()( YEXEyfyxfxj

jjji

i

X\Y 0 1 2 T

0 1/16

1/81/16

1/4

1 1/8 1/4 1/8 1/2

2 1/16

1/81/16

1/4

T 1/4 1/2 1/4 1.0

X\Y 0 1 2 T

0 1/8 1/8 0 1/4

1 1/8 1/4 1/8 1/2

2 0 1/8 1/8 1/4

T 1/4 1/2 1/4 1.0

1f 3f

0),( YXCov 4/1),( YXCov

X\Y 0 1 2 T

0 0 1/4 0 1/4

1 1/4 0 1/4 1/2

2 0 1/4 0 1/4

T 1/4 1/2 1/4 1.0

2f

0),( YXCov

X Y X Y f1 f2 f3

0 0 0 1/16 1/8 0

0 1 0 1/8 1/8 1/4

0 2 0 1/16 0 0

1 0 0 1/8 1/8 1/4

1 1 1 1/4 1/4 0

1 2 2 1/8 1/8 1/4

2 0 0 1/16 0 0

2 1 2 1/8 1/8 1/4

2 2 4 1/16 1/8 0

Total 1.0 1.0 1.0

E(XY) 1 5/4 1

Cov(X,Y)

0 1/4 0

YX

0),( YXCov

0),( YXCov

YX

X

),(2)()()( YXCovYVarXVarYXVar

)()(2))(())(( 22 YEXEYEXE

22 )]([)()()( WEWEWVarYXVar

YXW

)(2)()(])[()( 2222 XYEYEXEYXEWE

222 )]()([)]([)]([ YEXEYXEWE

XYYXW 2222

(X)

(0)

YX

)()()( YVarXVarYXVar

)()()( 22 YVarbXVarabYaXVar

)()()( YVarbXVarabYaXVar

0),( YXCov

8<

:

8<

:

X

Y)( YXVar

)(XVar

)(YVar

YX

(o)

(X)

)()()( YEXEYXE

)()()( 22 YEbXEabYaXE

)()()( YEbXEabYaXE

YX regardless of independence

YX

25)( XVar

)( YXVar

9)( YVar

?

a

a b

12)1(! nnn

How many ways to give an order to n people ?

How many ways to give an order to r people

selected from n people ? --- No order for (n-r) people.

Give an full order to n people, and disregard

the order of the last (n-r) people.

)!(

!

rn

nPnr

How many ways to separate n people into two

groups of r people and (n-r) people ?

Disregard also the order of selected r people.

)!(!

!

rnr

n

r

nC nr

X: the number of heads when we toss a fair coin twice

2,1,0,5.02

)( 2

xx

xf

X f

0 0.25

1 0.5

2 0.25

Total 1.0

Thank you !!