Bivariata

X

x X

i P h

f = f = n /n j = 1, . . . , k

.j ij .j

i=1

Y

y Y

j

P P P P

h k h k

f = f = f = 1

ij i. .j

i=1 j=1 i=1 j=1 Y

X f = n /n

i. i.

f = 3/20 = 0.15 f = 2/20 = 0.1 f = 1/20 = 0.05 f = 6/20 = 0.3

11 12 13 1.

f = 1/20 = 0.05 f = 4/20 = 0.2 f = 2/20 = 0.1 f = 7/20 = 0.35

21 22 23 2.

f = 1/20 = 0.05 f = 3/20 = 0.15 f = 3/20 = 0.15 f = 7 = 0.35

31 32 33 3.

f = n /n f = 5/20 = 0.25 f = 9/20 = 0.45 f = 6/20 = 0.3

.1 .2 .3

.j .j X Y

X X Y

X

X n i.

x n

1 1.

x n

i i.

x n

h h.

n X

n y Y

.j j

X y j

X y j

x n

1 1j

x n

i ij

x n

h hj

n .j

Y k k

Y k

X k

Y n , . . . , n

.1 .k

X k

X X Y

X k

Y X

X n

X y y y

1 j k

n n

n n

1j

x f =

11 1k 1.

1 1.

n n n n

.1 .j .k

n n

n n

ij

x f =

i1 ik i.

i i.

n n n n

.1 .j .k

n

n n n

hj

x f =

h1 hk h.

h h.

n n n n

.1 .j .k

X Y Y

X

Y

Y n .j

y n

1 .1

y n

j .j

y n

k .k

n

Y

x X

i Y x i

Y x i

y n

1 i1

y n

j ij

y n

k ik

n i.

X h h

X

Y X h

X n , . . . , n

i. h.

Y Y h

Y

Y x x x

1 i h

n

n

n n

y f =

11 i1 h1 .1

1 .1

n n n n

1. i. h.

n

n n n

1j ij hj .j

y f =

j .j

n n n n

1. i. h.

n n n n

y f =

1k ik hk .k

k .k

n n n n

1. i. h.

X Y

h Y x i = 1, . . . , h

i

k X y j = 1, . . . , k

j X

Y

X

Y X

(0; 15] (15; 30] (30; 45] Y

(0; 7] (7; 14] (14; 21] Y

X (0; 7] (7; 14] (14; 21] n i.

(0; 15]

(15; 30]

(30; 45]

n .j X Y

X

X n f = n /n

i. i. i.

(0; 15] 9/18 = 0.50

(15; 30] 7/18 = 0.38̄

(30; 45] 2/18 = 0.11̄

Y

Y n f = n /n

.j .j .j

(0; 7] 12/18 = 0.66̄

(7; 14] 5/18 = 0.27̄

(14; 21] 1/18 = 0.05̄ X Y

X Y

X (0; 7] (7; 14] (14; 21]

(0; 15] 6

(15; 30] 5

(30; 45] 1

X Y

X (0; 7] (7; 14] (14; 21]

(0; 15] 6/12 = 0.500

(15; 30] 5/12 = 0.166̄

(30; 45] 1/12 = 0.083̄ Y X

Y (0; 15] (15; 30] (30; 45]

(0; 7] 6

(7; 14] 3

(14; 21] 0

Y X

Y (0; 15] (15; 30] (30; 45]

(0; 7] 6/9 = 0. 6̄

(7; 14] 3/9 = 0. 3̄

(14; 21] X

Y Y

X (800; 1500] (1500; 2000] (2000; 3000] n i.

n .j Y X

Y f i.

(800; 1500] 15/30 = 0.5 45/90 = 0.5 40/80 = 0.5 100/200 = 0.5

(1500; 2000] 12/30 = 0.4 36/90 = 0.4 32/80 = 0.4 80/200 = 0.4

(2000; 3000] 3/30 = 0.1 9/90 = 0.1 8/80 = 0.1 20/200 = 0.1

Y X Y

X (800; 1500] (1500; 2000] (2000; 3000] f .j

15/100 = 0.15 12/80 = 0.15 3/20 = 0.15 30/200 = 0.15

45/100 = 0.45 36/80 = 0.45 9/20 = 0.45 90/200 = 0.45

40/100 = 0.40 32/80 = 0.40 8/20 = 0.40 80/200 = 0.40

X X Y

X

X Y

X

X 0 1 0 1 0 1 0 1

n n

1j

n n

1k

11 1.

n n

n n

.j .k

.1

B C B C B

C B C

B C B C B C B C

B C B C B C B C

n

B C B C B C B C

n

n n

ij

··· ···

= = = = =

ik

i1 i.

B C B C B C B C

n n

n n

.j

.1 .k

B C B C B C B C

@ A @ A @ A @ A

n n

n n

hj h.

h1 hk n

n n

n

.1 .k

.j

n n

ij i.

= , i = 1, . . . , h j = 1, . . . , k.

n n

.j

Y X

Y Y

n n

ij .j

= , i = 1, . . . , h j = 1, . . . , k.

n n

i. X Y

X Y Y X

·

n n

i. .j

n = i = 1, . . . , h j = 1, . . . , k

ij n

n

X Y ·

f = f f i = 1, . . . , h j = 1, . . . , k

ij i. .j ·

n n

i. .j

n̂ =

ij n

X Y

Y

X (18; 22] (22; 26] (27; 30] n i.

n .j ·

n n

i. .j

n̂ = i =

ij n

1, 2, 3 j = 1, 2, 3 Y

X (18; 22] (22; 26] (27; 30] n i.

· · ·

42 33 42 55 42 62

= 9.24 = 15.40 = 17.36

150 150 150

n .j X

Y 2

X

2

X

X Y

X Y X Y h = k

Y Y

X y y y n X y y y n

1 2 3 i. 1 2 3 i.

x n n x n n

1 11 11 1 11 11

x n n x n n

2 23 23 2 22 22

x n n x n n

3 32 32 3 33 33

n n n n n n n n n n

.j 11 32 23 .j 11 22 33 h>k

X Y Y

X Y X

Y

X y y y n

1 2 3 i.

x n n

1 11 11

x n n

2 22 22

x n n

3 33 33

x n n

4 43 43

n n n n n

.j 11 22 .3 k > h

Y X X

Y X Y

Y

X y y y y n

1 2 3 4 i.

x n n n

1 11 14 1.

x n n

2 22 22

x n n

3 33 33

n n n n n

.j 11 22 .3 X Y

n n̂ (i, j)

ij ij ⇥

h k

2

X

h k

X X 2 ·

(n n̂ ) n n

ij ij i. .j

2

X = , n̂ =

ij

n̂ n

ij

i=1 j=1 2

X

2

X

2

max X 2 ·

max X = n min{h 1; k 1}.

2

X !

h k

X X 2

n ij

2

X = n 1 .

·

n n

i. .j

i=1 j=1

P P 2

n

h k ij 1

2

X i=1 j=1 ·n

n

2 i. .j

X̃ = = ,

2

max X min{h 1; k 1}

2 2

 

X 0 X̃ 1

2

X̃ 2

X Y X̃

2

X

2

X

n n̂ (18; 22] (22; 26] (27; 30]

ij ij 2

(n n̂ )

ij ij

n̂ ij

2

X

2

(n n̂ )

ij ij (18; 22] (22; 26] (27; 30]

n̂ ij 2 2 2

0.76 /9.24 = 0.0625 ( 0.4) /15.4 = 0.0104 ( 0.36) /17.36 = 0.0075

2 · · ·

X = 0.0625 + 0.0104 + + 2.1793 = 12.1910

2

X 2 · · ·

max X = n min{h 1; k 1} = 150 min{2; 2} = 150 2 = 300,

2

X 12.1910

2

X̃ = = = 0.0406.

2

max X 300

2

X̃ = 0.0406

Y

X Y

Y

X 1 2 3 4 5 n i.

n .j 2 2 2

Y X X̃ = X / max X

2

X

2 · · ·

max X = n min{h 1; k 1} = 200 min{2; 2} = 200 2 = 400.

2

X

2

n 2

ij X

·n

n i. .j

2

n ij 1 2 3 4 5

·n

n i. .j 2 2 2

· · ·

10 /(70 60) = 0.024 30 /(70 85) = 0.151 30 /(70 55) = 0.234

!

h k

X X 2

n ij

2 ·

X = n 1 = 200 (0.024 + 0.151 + . . . + 0.023 1) = 24.6.

·

n n

i. .j

i=1 j=1 2

X 24.6

2

X̃ = = = 0.0615.

2

max X 400

2

X̃ = 0.0615

Y X

Y

X 1 2 3 4 5 Y

Y X 2

X Y

X

NO 2

X X Y X Y

X Y

Y X Y

Y X Y X

h Y x , . . . , x

1 h

Y

X Y

y y y n

1 j k i.

x n n n n

1 11 1j 1k 1.

x n n n n

i i1 ij ik i.

x n n n n

h h1 hj hk h.

n n n n n

.j .1 .j .k

• Y

Y P k ·

y n

j .j

j=1

ȳ = n

• Y

Y P P

k k

2 2

· ·

(y ȳ) n y n

j .j .j

j

j=1 j=1

2 2

s̃ = = ȳ

Y n n

2

ȳ s̃ n

y X

x n

i i.

• Y x i

Y x i P k ·

y n

j ij

j=1

ȳ =

i n i.

• Y x i

Y x i

P k 2 ·

(y ȳ ) n

j i ij

j=1

2

s̃ =

i n i.

2

ȳ|x s̃ n

i i.

i x X i =

i

1, . . . , h h Y

X h Y ȳ , . . . , ȳ

1 h

h X

X ȳ n

i i.

x ȳ n

1 1 1.

x ȳ n

2 i i.

x ȳ n

h h h.

n

Y P h ·

ȳ n

i i.

i=1 = ȳ

n

h Y

P h 2 ·

(ȳ ȳ) n

i i.

i=1

2

s̃ =

B n 21 2

h Y s̃ , . . . , s̃ h

h X

2

s̃ W P h 2 ·

s̃ n i.

i

i=1

2

s̃ =

W n Y

k h h k

X X X X

2 2 2

· · ·

(y ȳ) n = (ȳ ȳ) n + (y ȳ ) n

j .j i i. j i ij

j=1 i=1 i=1 j=1

h h

X X

2 2

· ·

= (ȳ ȳ) n + s̃ n

i i. i.

i

i=1 i=1

n

P P P

k 2 h h

·

(y ȳ) n 2 2

· ·

(ȳ ȳ) n s̃ n

j .j i i. i.

j=1 i

i=1 i=1

= + .

n n n

2 2 2

s̃ = s̃ + s̃

Y B W 2

⌘

2 2 2

s̃ = s̃ + s̃ Y

Y B W Y n

ȳ , . . . , ȳ Y

1 h

X 2

· · ·

ȳ = ȳ = = ȳ s̃ = 0

1 2 h B

2

s̃ = 0 ȳ = ȳ i = 1, . . . , h

i

B Y Y X

2

s̃ =0

B Y

2

X ⌘ 2

s̃ B

2

⌘ = 2

s̃ Y

2

 

0 ⌘ 1

2 2

• ⌘ = 0 s̃ = 0 Y X

B Y

2 2 2 2

• ⌘ = 1 s̃ = s̃ s̃ = 0

B Y W Y

X 2

• 0 < ⌘ < 1

2

⌘ Y

X X Y 2

⌘

X

Y X

Y

X Y

X X Y

Y P k ·

y n

j .j

j=1

ȳ = .

n ·

n y y n

Y .j j j .j

·

(0; 7] 3.5 12 = 42

·

(7; 14] 10.5 5 = 52.5

(14; 21] 17.5

ȳ = 112/18 = 6. 2̄

Y Y

y j

X 3.5 10.5 17.5 n i.

(0; 15]

(15; 30]

(30; 45] X

n X · · ·

Y ȳ i = 1, , h

i

P P P

3 3 3

· · ·

y n y n y n

j 1j j 2j j 3j

j=1 j=1 j=1

ȳ = ; ȳ = ; ȳ = .

1 2 3

n n n

1. 2. 3.

(0; 15] (15; 30] (30; 45]

· · ·

y n y n n y n n y n

j 1j j 1j 2j j 2j 3j j 3j

· ·

3.5 3.5 6 = 21 3.5 5 = 17.5

·

10.5 10.5 3 = 31.5

17.5 ȳ = 52.5/9 = 5.83̄, ȳ = 45.5/7 = 6.5, ȳ = 14/2 = 7.

1 2 3

2

⌘

2 2 2

⌘ s̃ s̃

B Y 2 ·

ȳ n ȳ n

i i. i.

i

2 ·

(0; 15] 5.83̄ (5.83̄) 9 = 305.9

2 ·

(15; 30] (6.5) 7 = 295.75

·

(30; 45] 49 2 = 98

P P

h h

2 2

· ·

(ȳ ȳ) n ȳ n 699.65

i i. i.

i

2 2 2

i=1 i=1

s̃ = = ȳ = (6.222) = 38.869 38.713 = 0.156;

B n n 18

P k 2 ·

y n .j

j

j=1

2 2

s̃ = ȳ ,

Y n 2 ·

y n y n

j .j .j

j

·

12.25 12 = 147

·

110.25 5 = 551.25

306.25

2 2

s̃ = 1004.5/18 (6. 2̄) = 55.805 38.713 = 17.092

Y

0.156

2

⌘ = = 0.009

17.092

X Y {(x , y ), (x , y ), . . . , (x , y )}

1 1 2 2 n n

X Y

X Y n X

Y (x , y )

i i

i = 1, . . . , n X Y

x y

i i

120 18

125 19

127 20

128 22