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: 4.1....................... 1 4.1.1............... 1 4.2........... 10 4.2.1............... 10 4.2.2..... 14 4.2.3................ 18 4.2.4................ 24 4.3...................... 26 4.3.1.............. 26 4.3.2........... 28 4.3.3. 30 Previous Next First Last Back Forward 2

4.3.4.............. 35 4.4....................... 40 Previous Next First Last Back Forward 1

4.1 4.1.1,.,..,.,..,.. 1. Previous Next First Last Back Forward 1

: ( ).. 2000.,,.,, 13,,.,,.,,,.. Example Example,,. Previous Next First Last Back Forward 2

10000. 100.,, 70%, 30%. 100, 70, 30,. Example Example., : Previous Next First Last Back Forward 3

.,. Example 1 2 3 4 800 1000 1200 1400 10 20 30 40 A B C D 3, 4 4 3 = 64..,?. Example, Previous Next First Last Back Forward 4

.. :. 2.,,,,.,,. Previous Next First Last Back Forward 5

a, 5 x 1, x 2,, x 5, ( ). a : (1) 5 x = 1 5 (x 1 + + x 5 ) a; (2) x 1, x 2,, x 5 x (1) x (2) x (5), x (3) a; (3) W = 1 2 (x (1) + x (5) ) a. x x (3), x (3) W.???,. Example Example.,.. Previous Next First Last Back Forward 6

100,. 1. 90 5000, 10 10,? Example Example (1) : x = (90 0.5 + 10 10)/100 = 1.45( ) :. 90% 5000,. (2) : 100 x 1, x 2,, x 100, x (1) x (2) x (100)., (x (50) + x (51) )/2 = 0.5( ) :.. Previous Next First Last Back Forward 7

3.,.,.,,,. ( ). :,. :,...,,, ( )., :,,,,.,. Previous Next First Last Back Forward 8

,.,.,,...,,. Previous Next First Last Back Forward 9

4.2 4.2.1. 10000,,,, 100. 10000,, 100.,.. Example Example :, Previous Next First Last Back Forward 10

.,,.,, ( ),,. 4.2.1 0, 1, 0 1. :,.,.,. 4.2.1, 10000 100,, 0.01. X : X = { 1 0, Previous Next First Last Back Forward 11

0 1, P (X = 1) = 0.01., X., X, :.. Definition,.,., X, X, F. F, f, f., 0 1. F, Previous Next First Last Back Forward 12

(i.i.d.) n X 1,, X n, X 1,, X n i.i.d. F (4.1) F f, X 1,, X n i.i.d. f (4.2) X F, X 1,, X n X, X 1,, X n i.i.d. X (4.3) (4.1) (4.1) (4.3).,., X Y, (X, Y ) F (x, y). Previous Next First Last Back Forward 13

4.2.2 1.. X = (X 1,, X n ), : X = (X 1,, X n ), X. Definition, X = {(x 1,, x 5 ) : 0 < x i <, i = 1, 2, 5}, X = {(x 1,, x 5 ) : < x i <, i = 1, 2, 5}.,,, 0. : Previous Next First Last Back Forward 14

,,. X = (5, 1, 9) 5 1 9. Example X = {(x 1, x 2, x 3 ) : x i = 0, 1, 2, 10, i = 1, 2, 3},. Example 1,, 5.2.3, 3 X = (X 1, X 2, X 3 ), 0 X i 10, i = 1, 2, 3,.,,,,.,, (X 1, X 2, X 3 ). Previous Next First Last Back Forward 15

,, ( )..,,,,.,,, ( ).,,. ( ),,. 2.,,., : Previous Next First Last Back Forward 16

(1).,.,. (2).., X 1, X 2,, X n. (X 1,, X n ). : F, X 1,, X n F n, (i) X 1,, X n, (ii) X 1,, X n, F, (X 1,, X n ),. Definition Previous Next First Last Back Forward 17

F, (X 1,, X n ), X 1,, X n : n F (x 1 ) F (x 2 ) F (x n ) = F (x i ) F f, f(x 1) f(x 2) f(x n) = n f(x i).,. 4.2.3,,.. Previous Next First Last Back Forward 18

, ( ),,. : N, M, N, M. n, M p = M/N. (1),,,, n.. (2),,, n.. Example Example : (1),, N 1/N, P (X i = 1) = M/N, P (X i = 0) = Previous Next First Last Back Forward 19

(N M)/N, ( ) a ( ) n a M N M P (X 1 = x 1,, X n = x n ) =, (4.4) N N x 1, x n 0 1, x i = a ( 0). (2),,, : n x i = a, P (X 1 = x 1, X 2 = x 2,, X n = x n ) = M N M 1 N 1 M a + 1 N a + 1 N M N a N M n + a + 1,(4.5) N n + 1 x 1, x n 0, 1, x i = a ( 0).,, X 1,, X n,,. Previous Next First Last Back Forward 20

, X 1,, X n,. n/n, (4.5) (4.4). n/n.,,.,. 4.2.3.,,,.. a, n, X 1,, X n, X 1,, X n. Example Example Previous Next First Last Back Forward 21

: X 1,, X n,, : (1),. X 1,, X n. (2),,, ( )., X 1,, X n., X 1,, X n n,. X 1,, X n, X 1. :,.., 0. X 1 ( a ) N(a, σ 2 ). Previous Next First Last Back Forward 22

X 1,, X n f(x 1,, x n ) = ( 2πσ) n exp{ 1 2σ 2 (x i a) 2 } (4.6), : (i) X 1,, X n i.i.d., (ii),.,,., 4.2.3, (4.6).,.. 4.2.3 N(a, σ 2 ), X 1,, X n, (4.6)., 4.2.3, a. Previous Next First Last Back Forward 23

σ 2, a σ 2. 4.2.4.,,..,,,.,.,. :. 4.2.3 ( ) N(a, σ 2 ), a σ 2, n X 1,, X n, a σ 2 Previous Next First Last Back Forward 24

, a 1.,., 100%,. 4.2.3, X = 1 n X i a, X a c, P ( X a > c), X a. : (1). (2), X N(a, σ 2 ) a, P ( X a > c). (3),. Previous Next First Last Back Forward 25

4.3.1 4.3.,,. :,., :,,. Definition : (1),. X Previous Next First Last Back Forward 26

N(a, σ 2 ), X 1,, X n X i.i.d., n Xi 2, a σ 2, Xi 2 /σ 2. n X i (X i a) (2),, ;,. ( ),,. (3),.,,. Previous Next First Last Back Forward 27

4.3.2 1. : X 1,, X n X, X = 1 n X i... 2. : X 1,, X n X, S 2 = 1 n 1 (X i X) 2,. S,. 3. : X 1,, X n F, a k = 1 n Xi k, k = 1, 2, Previous Next First Last Back Forward 28

k, k = 1, a 1 = X. m k = 1 (X i n X) k, k = 2, 3, k. 4. : X 1,, X n F, X (1) X (2) X (n), (X (1), X (2),, X (n) ), (X (1),, X (n) ). : (1) : { X( n+1 n m 1 = 2 ) 2 1 [X (4.7) 2 ( n 2 ) + X ( n 2 +1)] n.,, m 1/2. Previous Next First Last Back Forward 29

(2) : X (1) X (n).. 5. : F, F,, X 1,, X n, F : F n(x) = {X 1,, X n x }/n X 1,, X n. 4.3.3,. 1. X 1,, X n i.i.d. N(a, σ 2 ), c 1, c 2,, c n Previous Next First Last Back Forward 30

, T = c k X k k=1 ( N a c k, σ 2 k=1, c 1 = = c n = 1/n, T = 1 n n c 2 k k=1 ) X i = X, X N(a, σ 2/ n). 2.. 1. X 1, X 2,, X n i.i.d. N(a, σ 2 ), X = 1 n (X i X) 2, 1 n 1 (1) X N(a, 1 n σ2 ); X i S 2 = Previous Next First Last Back Forward 31

(2) (n 1)S 2 /σ 2 χ 2 n 1; (3) X S 2. : (1) 5.5.2. (2) 1 n 1 n 1 n a 21 a 22 a 2n A =... a n1 a n2 a nn ( Schmidt ), Y = AX, Y 1 = 1 n n X i = n X, Previous Next First Last Back Forward 32

(n 1)S 2 = Y 2 1 + + Y 2 n = X 2 1 + + X 2 n. (X i X) 2 = Xi 2 n X 2 = Y 2 i Y 2 1 = i=2 Y 2 i. (4.8)?? Y i N(µ i, σ 2 ), i = 2,, n. A µ i = a a ik = 1 na a ik = 0. (4.9) n k=1 k=1 Previous Next First Last Back Forward 33

[ n ] Cov(Y i, Y j ) = E[(Y i EY i )(Y j EY j )] = E a ik (X k a) a jl (X l a) = k=1 l=1 k=1 a ik a jl E[(X k a)(x l a)] = n { σ = σ 2 2 i = j, a ik a jk = 0 i j. k=1 k=1 l=1 l=1 a ik a jl δ kl σ 2 δ kl = 1, k = l; 0. Y 2,, Y n i.i.d. N(0, σ 2 ). Y i /σ N(0, 1), i = 2,, n, (4.8) (n 1)S 2 σ 2 = (Y i /σ) 2 χ 2 n 1. i=2 (3) (2) Y 1, Y 2,, Y n, S 2 Y 2,, Y n, X Y1, X S 2,. Previous Next First Last Back Forward 34

4.3.4. 1. X 1, X 2,, X n (i.i.d.) N(a, σ 2 ), n( X a) T = t n 1. S : 5.4.3 X N(a, σ 2 /n), n( X a)/σ N(0, 1). (n 1)S 2 /σ 2 χ 2 n 1, S 2 /σ 2 χ 2 n 1/(n 1), X S 2, n( X a)/σ n( X a) T = = t n 1. S2 /σ 2 S Previous Next First Last Back Forward 35

2. X 1, X 2,, X m i.i.d. N(a 1, σ1), 2 Y 1, Y 2,, Y n i.i.d. N(a 2, σ2), 2 σ1 2 = σ2 2 = σ 2, X 1, X 2,, X m Y 1, Y 2,, Y n, T = ( X Ȳ ) (a 1 a 2 ) S w mn n + m tn+m 2, (n + m 2)Sw 2 = (m 1)S1 2 + (n 1)S2, 2 S1 2 = 1 m (X i m 1 X) 2, S2 2 = 1 (Y j n 1 Ȳ )2. j=1 : 5.4.3 X N(a, σ 2 /m), Ȳ N(a 2, σ 2 /n), X Ȳ N ( a 1 a 2, ( 1 + 1 )σ2) = N ( a m n 1 a 2, n+m σ2). mn X Ȳ (a 1 a 2 ) mn N(0, 1). (4.10) σ m + n (m 1)S 2 1/σ 2 χ 2 m 1, (n 1)S 2 2/σ 2 χ 2 n 1, χ 2 (m 1)S 2 1 + (n 1)S 2 2 σ 2 χ 2 n+m 2. (4.11) Previous Next First Last Back Forward 36

(4.10) (4.11) ( X, Ȳ ) (S2 1, S 2 2), T = ( X Ȳ ) (a 1 a 2 ) σ = ( X Ȳ ) (a 1 a 2 ) S w mn n + m nm n + m / (m 1)S 2 1 + (n 1)S2 2 σ 2 (n + m 2) t n+m 2. 3. X 1, X 2,, X m i.i.d. N(a 1, σ 2 1), Y 1, Y 2,, Y n i.i.d. N(a 2, σ 2 2), X 1, X 2,, X m Y 1, Y 2,, Y n, F = S2 1 σ2 2 S2 2 σ1 2 S 2 1 S 2 2 2. F m 1,n 1, : 5.4.3 (m 1)S 2 X/σ 2 1 χ 2 m 1, (n 1)S 2 Y /σ 2 2 χ 2 n 1, Previous Next First Last Back Forward 37

, F F = (m 1)SX 2 / σ (m 1) 1 2 (n 1)S 2 Y σ 2 2 / (n 1) = S2 X S 2 Y σ2 2 σ 2 1 F m 1,n 1.. χ 2.. 4. X 1, X 2,, X n i.i.d. : f(x, λ) = λe λx I [x>0], 2λn X = 2λ X i χ 2 2n. Previous Next First Last Back Forward 38

: 2λX 1 χ 2 2. ( F (y) = P (2λX 1 < y) = P X 1 < y ) = 2λ 0 y 2λ λe λx dx, f(y) = F (y) = { 1 2 e y 2 y > 0 0 y 0. f(y) 2 χ 2, 2λX 1 χ 2 2. χ 2 (3), 2λX i χ 2 2, i = 1, 2,, n;, 2λ n X i χ 2 2n. Previous Next First Last Back Forward 39

4.4, : 1. ; 2. ;,. 3.,. 4..,.,,.. Previous Next First Last Back Forward 40