3.1 ( ) (Expectation) (Conditional Mean) (Median) Previous Next

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2 3.1 ( ) (Expectation) (Conditional Mean) (Median) Previous Next First Last Back Forward 1

3 ( ): ( ), 3. :, :,, :, : Previous Next First Last Back Forward 1

4 3.1 ( ) (Expectation) (Mean),., 100,, ,? Example Example :, 3/4, 1/4., 2 1,, = 150( ), Previous Next First Last Back Forward 2

5 , = 50( ). X, X ( 2 1 ),, X : 200 0, 3/4 1/4., X, X.,. Previous Next First Last Back Forward 3

6 : : ? Example Example N, : 8 0.3N N N = 9.3N : 8 0.2N N N = 9.1N, N, ,. Previous Next First Last Back Forward 4

7 ( ) :, X, P (X = x i ) = p i, i = 1, 2, i=1 x i p i < +, x i p i i=1 Definition X ( ), EX. x i p i = +, X ( ). i=1 Previous Next First Last Back Forward 5

8 , X f(x) (a, b), a < b, X X : 1. {x i }, a = x 0 < x 1 <... < x n = b, x i = x i x i X, {t i }, x i 1 < t i x i P (X = t i ) = p i = P (x i 1 < X x i ) f(t i ) x i 3. :( x i 0) EX = t i p i t i f(t i ) x i xf(x)dx := EX <, x i p i t i f(t i ) x i x f(x)dx <. Previous Next First Last Back Forward 6 R R

9 X f(x), x f(x)dx <, xf(x)dx X, EX. Definition X. x f(x)dx =, Previous Next First Last Back Forward 7

10 . 1. X B(n, p): EX = n n! k. k!(n k)! pk (1 p) n k k=0 n 1 (n 1)! = np i!(n 1 i)! pi (1 p) n 1 i = np. i=0 2. Poisson X P (λ): 3. X U[a, b]: EX = b a EX = λ. x 1 b a dx = a + b 2. Previous Next First Last Back Forward 8

11 4. X N(µ, σ 2 ): EX = = = µ. 5. X Exp(λ): EX = x e (x µ)2 2σ 2 dx 2πσ 1 (σy + µ). e y2 /2 dy 2π xλe λx dx = 1/λ. 6. X χ 2 n: 7. t X t n : EX = n. EX = 0. Previous Next First Last Back Forward 9

12 r.v. X ) P (X = ( 1) k 2 k k X = 1, k = 1, 2, 2k Example Example : ( 1) k 2 k k 1 2 = 1 k k = + k=1 X k=1 ( 1) k 2 k 1 k 2 = ( 1) k 1 k k = ln2. k=1 k=1 Previous Next First Last Back Forward 10

13 (Cauchy ) Example p(x) = 1 π(1 + x 2 ), x R, :. Example :,p(x), p(x)dx = 1 π x 2 dx = 1 π arctan x = 1, p(x) ( Cauchy ), x p(x)dx = 2 π Cauchy. # 0 x dx =, 1 + x2 Previous Next First Last Back Forward 11

14 ,. c 1, c 2,..., c n, E(c 1 X 1 + c 2 X c n X n ) = c 1 EX 1 + c 2 EX c n EX n,. X B(n, p), EX. Example Example : I i B(1, p), i = 1, 2,..., n, X = n i=1 I i EI i = p. EX = n i=1 EI i = np. Previous Next First Last Back Forward 12

15 2.,, E(X 1 X 2 X n ) = EX 1 EX 2 EX n,. 3. ( ) X, P (X = a i ) = p i, i = 1, 2,...,, f(x). { i Eg(X) = g(a i)p i, i g(a i) p i < ; + g(x)f(x)dx, + g(x) f(x)dx <. Previous Next First Last Back Forward 13

16 c, EcX = cex. Example Example X N(0, 1), Y = X Example Example : X N(0, 1), EX 2 = = 1., EY = EX = 2. + x 2 1. e x2 2 dx 2π Previous Next First Last Back Forward 14

17 X B(n, p), Y = X(n X). Example Example : X B(n, p), EX = np, EX 2 = V ar(x) + (EX) 2 = np(1 p) + n 2 p 2. EY = EX(n X) = ( ) n n k(n k) p k (1 p) n k k k=0 = nex EX 2 = n(n 1)p(1 p). Previous Next First Last Back Forward 15

18 20, 10,., X, EX. : Example Example Y i = { 1, i 0, i i = 1,, 20. X = 20 i=1 Y i, EX = = EY i = P ( i ) i=1 i=1 20 i=1 [ ] = Previous Next First Last Back Forward 16

19 3.1.3 (Conditional Mean),,. X x, Y, E(Y X = x), E(Y x). X Y, (X, Y ), X = x, Y P (Y = a i X = x) = p i, i = 1, 2,..., (X, Y ), X = x, Y f(y x). Definition { + E(Y X = x) = yf(y x)dy, i a ip i, (X, Y ) ; (X, Y ). Previous Next First Last Back Forward 17

20 . (X, Y ) N(a, b, σ 2 1, σ 2 2, ρ), E(Y X = x). Example Example : Y X = x N(b + ρ σ 2 σ 1 (x a), (1 ρ 2 )σ 2 2), E(Y X = x) = b + ρ σ 2 σ 1 (x a). [ ]: E(Y X = x) x, x X, E(Y X). : Previous Next First Last Back Forward 18

21 1 (Law of total expectation). X, Y. EX = E{E[X Y ]} [ ] :. X p.d.f f(x), Y p.d.f p(y), X Y = y p.d.f q(x y). EX = = xf(x)dx = x xq(x y)dxp(y)dy = = E{E[X Y ]} q(x y)p(y)dydx E[X Y = y]p(y)dy [ ]: g(x), Eg(X) = E{E[g(X) Y ]}. : h(x) = E(Y X = x), Eh(X), EY. Previous Next First Last Back Forward 19

22 3,. 3 ; 2, 5 ; 3, 7. 3,. Example Example : X, Y 3, Y 1/3 1, 2, 3. EX = E[E(X Y )] = 3 E(X Y = i)p (Y = i) i=1 E(X Y = 1) = 3, E(X Y = 2) = 5 + EX, Previous Next First Last Back Forward 20

23 E(X Y = 3) = 7 + EX, EX = 15. EX = 1 [ EX EX] 3 (X, Y ) N(a, b, σ 2 1, σ 2 2, ρ), EXY. : Example Example E(XY X = x) = xe(y X = x) = x(b + ρ σ 2 σ 1 (x a)); EXY = E(bX + ρ σ 2 σ 1 X 2 ρ σ 2 σ 1 ax) = ab + ρ σ 2 (a 2 + σ1) 2 ρ σ 2 a 2 σ 1 σ 1 = ab + ρσ 1 σ 2. Previous Next First Last Back Forward 21

24 3.1.4 (Median), X,,.,.. m X, P (X m) 1 2, P (X m) 1 2. Definition, m X : Previous Next First Last Back Forward 22

25 m, m,, m,.., :,,,,, : 1.,., E(X 1 + X 2 ) = EX 1 + EX 2, X 1 + X 2 X 1, X 2,, ; 2.,. Previous Next First Last Back Forward 23

26 X B(1, 1 ), X. 2 Example Example : X 0, x 0 1 F (x) = 2 1, x 1 (0,1) X,. p : 0 < p < 1 µ p X p P (X µ p ) p, P (X µ p ) 1 p. Definition Previous Next First Last Back Forward 24

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: p Previous Next First Last Back Forward 1 7-2: : 7.2......... 1 7.2.1....... 1 7.2.2......... 13 7.2.3................ 18 7.2.4 0-1 p.. 19 7.2.5.... 21 Previous Next First Last Back Forward 1 7.2 :, (0-1 ). 7.2.1, X N(µ, σ 2 ), < µ 0;