(I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 Typeset by Akio Namba usig Powerdot. 2 / 47

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1 4 Typeset by Akio Namba usig Powerdot. / 47

2 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 Typeset by Akio Namba usig Powerdot. 2 / 47

3 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 (radom variable): :, 2, 3, 4, 5, 6 6 : x, x 2,, x X X x i P X x i p i X X x x 2 x P X x i p p 2 p p p 2 p i x x 2 x i Typeset by Akio Namba usig Powerdot. 3 / 47

4 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 X x i f x i P X x i p i X (probability fuctio) 2. f x i 0,, 2, () 2. f x i ( ) X P X x i f x i, x 6, 2,..., Typeset by Akio Namba usig Powerdot. 4 / 47

5 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 X x F x P X x r r p i f x i r x r x x r (distributio fuctio) p 2 p p r p r x x 2 x r x x r F 0, F Typeset by Akio Namba usig Powerdot. 5 / 47

6 (I) (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : H H c 3 X X H H H H H H c H H c H H H c H c H c H H H c H H c H c H c H H c H c H c Typeset by Akio Namba usig Powerdot. 6 / 47

7 (II) (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : X H H H H H H c H H c H H H c H c H c H H H c H H c H c H c H H c H c H c P X P X P X P X Typeset by Akio Namba usig Powerdot. 7 / 47

8 2 (I) (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) 4.2 () A p A c q p (3 ) A x f x P X x C x p x q x! x! x! px q x!! 2 0! : 2 Typeset by Akio Namba usig Powerdot. 8 / 47

9 2 (II) (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 C x x 3 3! 3C 0 0!3! 3! 3C 2 2!! C x C x 3!, 3C!2! 3! 3, 3C 3 3!0! 4.2 x f x P X x 3C x 0.3 x 0.7 x 3, Typeset by Akio Namba usig Powerdot. 9 / 47

10 2 (III) (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) X f x P X x C x p x q x! x! x! px q x X 2 (Biomial Distiributio) X B, p : 2 Typeset by Akio Namba usig Powerdot. 0 / 47

11 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : () (probability desity fuctio): P a X b a b f x dx : 2 f x X Typeset by Akio Namba usig Powerdot. / 47

12 (I) (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) P a X b a b f x dx a X a, b X b f x X : 2 P a X b b f x dx a Typeset by Akio Namba usig Powerdot. 2 / 47

13 (II) (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 f x 0, f x dx X a X a P X a P a X a a a P a X b P a X b P a X b P a X b f x dx 0 Typeset by Akio Namba usig Powerdot. 3 / 47

14 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) X x F x P X x x f t dt P a X b F b F a b f x dx a f x dx : 2 a b f x dx Typeset by Akio Namba usig Powerdot. 4 / 47

15 (II) (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) F 0, F : 2 Typeset by Akio Namba usig Powerdot. 5 / 47

16 (I) (II) (I) : 2 Typeset by Akio Namba usig Powerdot. 6 / 47

17 (I) (I) (II) (I) : ) ( Typeset by Akio Namba usig Powerdot. 7 / 47

18 (II) (I) (II) (I) : (expectatio, ) E X x i p i x i f x i E X xf x dx Typeset by Akio Namba usig Powerdot. 8 / 47

19 (I) (II) (I) : 2 4. a, b E ax b ae X b ( ): E ax b a ae X ax i b f x i ax i f x i bf x i x i f x i b b f x i Typeset by Akio Namba usig Powerdot. 9 / 47

20 (I) (II) (I) : 2 X (Variace) V X E X µ 2, µ E X x i µ 2 f x i () x µ 2 f x dx () σ X V X E X µ 2 Typeset by Akio Namba usig Powerdot. 20 / 47

21 (I) (I) (II) (I) 4.2 µ E X V X E X 2 µ 2 ( ): V X E X µ 2 x i µ 2 f x i : 2 x 2 i 2µx i µ 2 f x i x 2 i f x i 2µ x i f x i µ 2 f x i E X 2 2µE X µ 2 E X 2 µ 2 Typeset by Akio Namba usig Powerdot. 2 / 47

22 (I) (II) (I) : a, b V ax b a 2 V X : E ax b ae X b aµ b ( 4.) V ax b E ax b aµ b 2 E a X µ 2 E a 2 X µ 2 a 2 E X µ 2 ( 4. ) a 2 V X Typeset by Akio Namba usig Powerdot. 22 / 47

23 (I) (II) (I) X µ E X σ σ X z X µ σ X () (stadardized) : 2 Typeset by Akio Namba usig Powerdot. 23 / 47

24 (I) (II) (I) : z E z 0, V z : a σ, b µ σ z ax b 4. E z ae X b E X µ σ µ σ µ σ σ V z a 2 V X σ 2 σ 2 Typeset by Akio Namba usig Powerdot. 24 / 47

25 (I) (II) (I) a k E X a k a k (momet) : E X :0() V X E X E X 2 : E X 2 : 2 Typeset by Akio Namba usig Powerdot. 25 / 47

26 (I) (II) (I) : 2 k m k E X E X k γ m 4 m 2 2, γ 2 m 3 m γ (kurtosis) γ 2 (skewess) Typeset by Akio Namba usig Powerdot. 26 / 47

27 (I) 4.9 () 4.9 () : 2 Typeset by Akio Namba usig Powerdot. 27 / 47

28 (I) (I) 4.9 () 4.9 () : 2 X, Y X Y (X Y j ) X i Y j (i, j, 2,, 6) P X i, Y j P X i P Y j 36 X Y (joit probability distributio) : 2 Typeset by Akio Namba usig Powerdot. 28 / 47

29 (II) (I) 4.9 () 4.9 () f x i, y j P X x i, Y y j p ij, 2,,, j, 2,, m X, Y X Y 4.6 y y 2 y m x p p 2 p m p x 2 p 2 p 22 p 2m p x p p 2 p m p p p 2 p m : 2 Typeset by Akio Namba usig Powerdot. 29 / 47

30 (II) (I) 4.9 () 4.9 () p i m i X Y 4.6 y y 2 y m x p p 2 p m p x 2 p 2 p 22 p 2m p x p p 2 p m p p p 2 p m p ij Y X x i X (margial distributio) p j Y : 2 Typeset by Akio Namba usig Powerdot. 30 / 47

31 (II) (I) 4.9 () 4.9 () f x i P X x i p i m j f y j P X y j p j X, Y m j p ij p i m j p j 2 X, Y p ij p ij : 2 Typeset by Akio Namba usig Powerdot. 3 / 47

32 (I) 4.9 () 4.9 () Y Y y j X x i f x i y j P X x i Y y j P X x i, Y y j P Y Y j () f x i, y j f y j f x i y j Y y j X x i : 2 Typeset by Akio Namba usig Powerdot. 32 / 47

33 (I) 4.9 () 4.9 () : 2 X x i Y y j P X x i, Y y j P X x i P Y y j f x, y f x, f y p ij, p i, p j f x i, y j f x i f y j, p ij p i p j i, j X Y () f x, y f x, f y f x, y f x f y X Y () Typeset by Akio Namba usig Powerdot. 33 / 47

34 (I) 4.9 () 4.9 () : 2 X, Y 4.6 X () E X m j x i x i p ij m x i p i Y j p ij Typeset by Akio Namba usig Powerdot. 34 / 47

35 (I) 4.9 () 4.9 () 4.5 X, Y : E X E X Y E X E Y Y E X m j m j x i y j p ij x i p ij E Y m j y j p ij : 2 Typeset by Akio Namba usig Powerdot. 35 / 47

36 (I) 4.9 () 4.9 () : X Y : E XY E XY m j m j x i p i E X E Y x i y j p ij x i y j p i p j m j E X E Y y j p j ( ) Typeset by Akio Namba usig Powerdot. 36 / 47

37 (I) 4.9 () 4.9 () V X E X E X 2 m j x i E X 2 p ij x i E X 2 p i V Y V X, V Y,, σ X ( σ X ), σ Y ( σ Y ) : 2 Typeset by Akio Namba usig Powerdot. 37 / 47

38 (I) 4.9 () 4.9 () (covariace) Cov X, Y E X E X Y E Y m j X x i E X y j E Y p ij Y : 2 Typeset by Akio Namba usig Powerdot. 38 / 47

39 (I) 4.9 () 4.9 () : Cov X, Y E XY E X E Y : Cov X, Y m j m j x i E X y j E Y p ij x i y j x i E Y E X y j E X E Y p ij E XY E X E Y E X E Y E X E Y E XY E X E Y X Y 4.6 E XY E X E Y Cov X, Y 0 Cov X, Y 0 X Y Typeset by Akio Namba usig Powerdot. 39 / 47

40 (I) 4.9 () 4.9 () (correlatio coefficiet) ρ X, Y Cov X, Y σ X σ Y X Y ρ X, Y 0 ρ X, Y 0 X Y : 2 Typeset by Akio Namba usig Powerdot. 40 / 47

41 4.8 ρ X, Y 0 (I) 4.9 () 4.9 () : 2 : V X Y V X V Y V X Y E X Y E X E Y E X E X Y E Y E X E X 2 Y E y 2 2 X E X Y E Y E X E X 2 E Y E Y 2 2E X E X Y E Y V X V Y 2E X E X Y E Y ρ X, Y 0 E X E X Y E Y 0 V X Y V X V Y. 2 2 Typeset by Akio Namba usig Powerdot. 4 / 47

42 (I) 4.9 () 4.9 () : 2 4.5(E X Y E X E Y ) 4.8(ρ X, Y 0 V X Y V X V Y ) 4.9 X, X 2,, X µ V X i σ 2 E X i µ, V X i σ 2,, 2,, X E X µ, V X X i σ 2 Typeset by Akio Namba usig Powerdot. 42 / 47

43 4.9 () (I) 4.9 () 4.9 () : 2 : 4.5 E X E E µ µ X i X i E X i µ ( 4.5 ) Typeset by Akio Namba usig Powerdot. 43 / 47

44 4.9 () (I) 4.9 () 4.9 () : V X V 2 V 2 2 σ2 σ 2 X i V X i X i ( 4.3 ) (X i 4.8 ) Typeset by Akio Namba usig Powerdot. 44 / 47

45 : : 2 Typeset by Akio Namba usig Powerdot. 45 / 47

46 2 : X B, p f x C x p x q x! x! x! px q x, x 0,,..., E X V X x 0 x 0 xf x p x E X 2 f x pq q p Typeset by Akio Namba usig Powerdot. 46 / 47

47 2 : A p A c q p X i i A A c 0 X X i X A X B, p P X p, P X i 0 q E X p 0 q p E Xi 2 2 p 0 2 q p V X i E Xi 2 2 E X i p p 2 pq 4.5 E X 4.8 V X E X i p V X i pq Typeset by Akio Namba usig Powerdot. 47 / 47

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