ใบงานเรื่อง ตัวแปรสุ่ม

1.
2.

-
-
3.
-
-

13

S

H

T

S HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

n(s) = 8 E3 3
E0, E1 , E 2

P E0 1 P E1 3
P E2 8 P E3 8
3 1
8 8

{0, 1, 2, 3} 3

(6 XS

1

X(HHH ) 3 X(HHT ) 2
X(HTH ) 2 X(HTT ) 1
X(THH ) 2 X(THT ) 1
X(TTH ) 1 X(TTT ) 0

X

(random variable)

X, Y, Z

x, y, z x X

13

x {0,1,2, 3} x
X=x
P(X 0) P (E 0 ) 0
P(X 1) P(E1) 1
P(X 2) P (E2 )
P(X 3) P (E 3 ) 3

1. (discrete random variable)

1

{2, 3, 4, 5, 6, 7, 8, 9,10,11,12}

(6

11 0 1
1
{0,1}

{1, 2, 3,...}

2. (continuous random variable)

6

[150,190]
[1, 6]

[0, ]

1 X1 50
100
X2
3
3 X3

X4

(6

5 X5 -
6 X6

(
(GB)

(
()
(
(
4

(

(6 4

6 50

() 6
0
1 17
4
3 1
4

1X

x x {0,1, 2, 3, 4}

X (probability distribution)

(6 5

x 01234

P(X x)

X 13
X

(6 6

Y Y

1

(6 7

01 3 4 5 6 7 8 9 10
01 5638733

X X

100 01 3456

Y 41 17 14 4 3 0 1

Y

(6 8

3. Z

Z

(6 9

(expected value) X
X
X

XX nn x x iPP( (XX xix)i )
i1
1 i

i

n X
x1, x 2 , x 3 ,..., x n

3 XX

XX nn XX)2)2PP(X(X xxi )i )

2 (x(xi i
X
i i1 1
n
x1, x 2 , x 3 ,..., x n X
X

X

4 6 50
1
X

06
1

17
34
41

(6 10

5
–7

6X 1

X

(6 11

7Y

Y

(6

01 3 4 5 6 7 8 9 10
01 5638733

X X

(6 13

100 01 3456

Y 41 17 14 4 3 0 1

Y

(6 14

Z
Z

(6 15

6 3 11

X

1) X

X

(6 16

2, 000, 000
2, 000, 000

50, 000

(

60

01 3

47 4 6 3

11 1

(6 17

4 X x1, x 2 , x 3 ,..., x n
X
X

(discrete uniform distribution)

P(PX(X xi )xi ) 1 1 i i {1,{21,,32,,.3.,..,.n.,}n}
n n

8 11 X

X

X

(6 18

91 X
500 X

1)

X

3 150 1

(6 19

1 X1 1
1 10
X2 80

3 X3

4 X4

01 34

16 16 16 16 16

(6

X X
X
10

100

(6

1

a

0a 5 Z

1 a5 Z

aZ

3 a1

(6

5 10
500
300

(6

5 (binomial distribution) n

X p
1-p

Note! -n p X

X B(n, p) p 1

n 1
(Bernoulli trial)
-

1n p 0p1
3
q (1 - p)
(6

1

X

P(X x) n px (q)n x x 0, 1, 2,..., n
x (1 p)

X np
X npq

n
p
q

10 X B(8, 0.7) X

P(X 4)

P(X 6)

(6

P(X 5)
P(3 X 7)

11 X B(5, 0.4)

P(X 1)

P(X 4)
P(X 4)

(6

P(1 X 4)

12 X 5 17
1
X

55

(6

3X

4X

(6

13
1)

(6

14 1
5

1)

) 30

(6

X B(6, 0.3) 31
P(X 2)

P(X 2)

P(X 2)
P(2 X 5)

(6

X X 6

1) X

)

3) 3

(6

4 X
5
18
3 Y
1) Y

(6 33

3 5
4) 8
4
Y

9
10

(6 34

5
5

1

6 35

(6

3
1

3 36

(6

7
9

1 4
3
(6
37

36
4

density curve)

X y f (x) X

(6 (probability density function)

38

Note! f (x) X

- f (x) 0 x X
-
1

X

y fx X 13

P1 X 3 3

f (x) dx

1

(6 39

Xa X P(X a) 0
aa

[a,b ]
(a,b)

Pa X b Pa X b
ab X

PX a PX a PX a PX a
a
X

08

7
–

(6 40

6
(normal distribution)

X

11x x 22

ff xx 11 ee 22

22 xx

X

X normal curve)

1X

XX X

3

(6 41

X 2 2
X
X N ,2 X

2

(6

7 (Standard normal distribution)

0 ( 0) 1 ( 1)

Z

f (fz()z) 1 1 e z2 z2 zz
2 2 2
e2

0 1

Standard normal curve)

random variable) Standard normal

Z z
z PZ z
(6
43

15 Z 44

PZ 2
P Z 1.29
P 1.27 Z 0.45

(6

16 Z 45

P Z 1.66

P Z 1.44

P 2.45 Z 0

) P 0 Z 1.88

(6

2

X

Z

ZX

Z z0 z1
Pa X b P a Zb

a,b X a b

17 X N (3.5, 4) P(2.4 X 5.2)

(6 46

X Z
ZX
) P( 1 Z 1)
P( X P(Z 1) P(Z
0.8413 0.1587
0.6826 1)

X [, ]

68

(6 47

2X 2) P( 2 Z 2)
P(Z 2) P(Z
0.9772 0.0228 2)
0.9544

X [ 2, 2] 9544
22

(6 48

P( 3 X 3) P( 3 Z 3)
P(Z 3) P(Z
0.9987 0.0013 3)
0.9974

X [ 3, 3] 9
33
9

(6 49