Download - Probability & Independence
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
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{(
Bet $100 + “all the amount you lose” every time.
Think coin tossing game.
Then you will win at least a time, then stop there.
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.
(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
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
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
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
),(2)()()( YXCovYVarXVarYXVar
)()(2))(())(( 22 YEXEYEXE
22 )]([)()()( WEWEWVarYXVar
YXW
)(2)()(])[()( 2222 XYEYEXEYXEWE
222 )]()([)]([)]([ YEXEYXEWE
XYYXW 2222
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