ST2334 complete notes PDF

Preview

Summary

Download ST2334 complete notes PDF

Description

S S = {1, 2, 3, 4, 5, 6}

S = {even,odd}

S = {(H, H ), (H , T ), (T , H ), (T , T )}

(H , T )

(T , H )

(H, H )

A = {t : 0 ≤ t < 5}

∅

A′

A

S

A∩B = ∅

B

A∪B n ⋃i=1 Ai

A

A

B

= A 1 ∪ A 2 ∪ ⋯ ∪ An ST2334 Notes by Hanming Zhu

A∩B

A

B

⋂ni=1 Ai = A1 ∩ A 2 ∩ ⋯ ∩ An

A ∩ A′ = ∅ A∩∅=∅ A ∪ A′ = S (A ′) ′ = A (A ∩ B)′ = A ′ ∪ B ′ (A ∪ B)′ = A ′ ∩ B ′ A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) A ∪ B = A ∪ (B ∩ A′ ) A = (A ∩ B) ∪ (A ∩ B ′ )

(A 1 ∪ A2 ∪ ⋯ ∪ An )′ = A′1 ∩ A ′2 ∩ ⋯ ∩ A′n (A 1 ∩ A2 ∩ ⋯ ∩ An )′ = A′1 ∪ A ′2 ∪ ⋯ ∪ A′n

A⊂B A⊂B

A B⊂A

B

A=B

n1

n2

n1 n2 n1 n2 ⋯ nk

k

n1

n2 n 1 + n2

n1 + n2 + ⋯ + nk

k

0, 1, 2, 5, 6, 9 0 5 × 4 = 20 0 4 × 4 × 2 = 32

r

0

r≤n

n n

r

nP r

=

n! (n−r)!

(n − 1)!

ST2334 Notes by Hanming Zhu

nk

n1 + n2 + ⋯ + nk = n

k

r

nP n1 ,n2 ,⋯ ,nk

=

n! n1 !n2 !⋯nk !

n n

(nr ) = n C r =

r

n! r!(n−r)!

n−1 n−1 n )for1 ≤ r ≤ n )+ ( ( ) =( r−1 r r

(61) ×(15) 30 = 55 ( 112 )

1/n

n

f A = nAn

A

n

E Pr(A) = limn→∞ fA

0 ≤ Pr(A) ≤ 1 Pr(S) = 1 Ai ∩ A j = ∅

A 1, A 2 , ⋯

i =j

∞

∞

i=1

i=1

Pr(⋃ Ai ) = ∑ Pr(Ai )

Pr(∅) = 0 n

n

Pr( ⋃i=1 Ai ) = ∑i=1 Pr(A i )

A 1, A 2 , ⋯ , A n A Pr(A ′) = 1 − Pr(A) A

B Pr(A) = Pr(A ∩ B) + Pr(A ∪ B′ )

A

B Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B) A, B, C Pr(A ∪ B ∪ C) = Pr(A) + Pr(B ) + Pr(C) − Pr(A ∩ B ) − Pr(A ∩ C ) − Pr(B ∩ C ) + Pr(A ∩ B ∩ C)

A⊂B

Pr(A) ≤ Pr(B)

n

n−1

n

n−2 n−1

n

Pr(A1 ∪ A2 ∪ ⋯ ∪ An ) = ∑ Pr(Ai ) − ∑ ∑ Pr(A i ∩ Aj ) + ∑ ∑ ∑ Pr(A i ∩ Aj ∩ A k ) − ⋯ + (−1)n+1 Pr(A1 ∩ A2 ∩ ⋯ ∩ An ) i=1

i=1 j=i+1

i=1 j=i+1 k=j+1

pn = Pr(A) = 1 − qn

ST2334 Notes by Hanming Zhu

1 − ( 364 )n ≥ 0.5 365

n

n≥

Pr(A∣B) =

log(0.5) = 252.7 log( 364 ) 365

Pr(A ∩ B) , ifPr(A) =0 Pr(B )

Pr(A ∩ B) = Pr(A)Pr(B∣A) = Pr(B )Pr(A∣B ) Pr(A ∩ B ∩ C) = Pr(A)Pr(B∣A)Pr(C∣A ∩ B )

n

n

i=1

i=1

Pr(B) = ∑ Pr(B ∩ A i) = ∑ Pr(Ai )Pr(B∣Ai ) A1 , ⋯ , An

A1 , A2 , ⋯ , An

S Pr(Ak ∣B) =

A

Pr(A ∩ B) = Pr(A)Pr(B)

B

Pr(B∣A) = Pr(B) A

B

A

B

Pr(A∣B) = Pr(A) Pr(A), Pr(B) > 0

∅

S A⊂B

A

Pr(Ak )Pr(B∣Ak ) n ∑i=1 Pr(Ai )Pr(B∣Ai )

A

B=S

B

B

A

B ′ A′

B A′

B′

n A 1, A 2 , ⋯ , A n

Pr(Ai ∩ Aj ) = Pr(A i )Pr(Aj )

i =j

i, j = 1, ⋯ , n

ST2334 Notes by Hanming Zhu

n {Ai1 , Ai2 ⋯ , Aik }

A1 , A2 , ⋯ , An

Pr(A i1 ∩ Ai 2 ∩ ⋯ ∩ Aik ) = Pr(Ai 1 )Pr(Ai2 ) ⋯ Pr(Aik )

s∈S

X R X = {x∣x = X (s), s ∈ S }

B

RX

A = {s ∈ S∣X(s) ∈ B}

A

B ⊂ RX Pr(A) = Pr(B)

B

A

X

X

f (x)

X (xi , f (xi )) f (xi ) ≥ 0 ∑∞ i=1 f (xi

f (x) X

xi ) =1

1

RX

X

X

f (x) ST2334 Notes by Hanming Zhu

f (x) ≥ 0

x ∈ RX f (x) = 0

∫R f (x)dx = 1 X

∞ ∫∞

f (x)dx = 1

f (x) = 0

(c, d) ⊂ RX , c < d Pr(c ≤ X ≤ d) = Pr(X = x0 ) =

x ∫x00

Pr(A) = 0

x∈ / RX d ∫c

x

A=∅

RX

f (x)dx

f (x)dx = 0 X

Pr(c ≤ X ≤ d) = Pr(c ≤ X < d) = Pr(c < X ≤ d) = Pr(c < X < d)

X F (x)

X F (x) = Pr(X ≤ x)

F (x) = ∑ f (t) = ∑ Pr(X = t) t≤x

t≤x

a ≤ b Pr(a ≤ B ≤ b) = Pr(X ≤ b) − Pr(X < a) = F (b) − F (a− ) a−

X

a

F (x) = ∫

x

f (t)dt

−∞

f (x) =

dF (x) dx

a ≤ b Pr(a ≤ X ≤ b) = Pr(a < X ≤ b) = F (b) − F (a)

0, ⎧ ⎪ ⎪ ⎪ 0.3, F (x) = ⎨ 0.9, ⎪ ⎪ ⎩ ⎪ 1,

X

E(X)

ifx < 0, if0 ≤ x < 1, if1 ≤ x < 2, if2 ≤ x.

μX

μX = E(X) = ∑ xi fX (xi ) = ∑ xfX (x) x

i

μX = E(X) = ∫

∞

xf X (x)dx

−∞

g(X)

X

f X (x)

E[g(X)] = ∑x g(x)f X (x) X ∞

E[g(X)] = ∫−∞ g(x)fX (x)dx X ST2334 Notes by Hanming Zhu

g(x) = (x − μX )2 σ2X = V (X) = E [(X − μX )2 ] = {

ifX isdiscrete, ∑x(x − μ X ) 2f X (x), ∞ ∫−∞(x − μX )2 fX (x)dx, ifX iscontinuous.

σX =

V (X )

V (X) = E(X2 ) − [E(X)]2

g(x) = xk

k

E(aX + b) = aE (X) + b 2

V (X) = E(X ) − [E(X)]

a

X

E(Xk )

b

2

V (aX + b) = a 2V (X)

X X

E(X)

V (X) V (X) = σ 2

E(X) = μ

k Pr(∣X − μ∣ ≥ kσ) ≤

1 k2

Pr(∣X − μ∣ < kσ) ≥ 1 − σ2

σ

(X, Y )

1 k2

X, Y

s∈S

(X, Y )

RX,Y = {(x, y )∣x = X (s), y = Y (s), s ∈ S } n

(X, Y )

(X (s), Y (s))

(X, Y )

(xi , yj ) fX,Y (xi , yj ) ≥ 0

R2

(X (s), Y (s))

fX,Y (xi , yj )

Pr(X = xi , Y = y j )

(xi , yj ) ∈ R X,Y

∞ ∞ ∞ ∑∞ i=1 ∑j=1 f X,Y (xi , yj ) = ∑i=1 ∑j=1 Pr(X = x i, Y = yj ) = 1

fX,Y

(X, Y )

ST2334 Notes by Hanming Zhu

fX,Y (x, y) fX,Y (x, y) ≥ 0 ∬(x,y)∈R

X,Y

(x, y) ∈ R X,Y ∞

∞

∫∞ ∫∞ f X,Y (x, y )dxdy = 1

fX,Y (x, y )dxdy = 1

fX (x) = ∑y f X,Y (x, y ) fY (y) = ∑x f X,Y (x, y ) ∞

fX (x) = ∫−∞ fX,Y (x, y )dy ∞

fY (y) = ∫−∞ fX,Y (x, y )dx

X=x

Y

fY ∣X (y∣x) =

f X,Y (x,y) fX (x) , iffX

(x) > 0

x

X

Y =y

X

fY ∣X (y∣x) ≥ 0

f X∣Y (x∣y) ≥ 0

fX (x) > 0 f X,Y (x, y) = fY ∣X (y∣x)fX (x)

X

fY (y) > 0 fX,Y (x, y) = fX∣Y (x∣y )fY (y )

fX,Y (x, y) = fX (x)fY (y )

Y

x, y

n

f X (x) fX (x) > 0

x ∈ A1

fY (y) > 0

f Y (y ) y ∈ A2

fX (x)fY (y) > 0

(x, y) ∈ A1 × A2

forDiscreteRV’s, ∑ ∑ g(x, y )f X,Y (x, y ), E[g(X, Y )] = { ∞ x ∞y ∫−∞ ∫−∞ g(x, y )fX,Y (x, y )dxdy , forCont.RV’s.

g(X, Y ) = (X − μX )(Y − μ Y )

μ X = E(x)

Cov(X, Y ) = E[(X − μX )(Y − μ Y )] Cov(X, Y ) = E(XY ) − μ X μY X

Cov(X, Y ) = 0

Y

Cov(X, Y ) = 0

Cov (aX + b, cY + d) = acCov (X, Y ) V (aX + bY ) = a2 V (X) + b2 V (Y ) + 2abCov (X, Y )

X

Y

Cor(X, Y )

ρX,Y

ρ

ρX,Y =

Cov(X,Y ) V (X) V (Y )

ST2334 Notes by Hanming Zhu

−1 ≤ ρX,Y ≤ 1 X

ρ X,Y X

ρX,Y = 0

Y

Y

ρX,Y = 0

x 1, x 2, ⋯ , x k f X (x) = k1′ x = x1 , x2 , ⋯ , xk

X μ = E(X) = ∑ki=1 x i 1k =

X

k xi ∑i=1

1 k

k

σ 2 = V (X) = ∑allx(x − μ)2 fX (x) = k1 ∑i=1 (xi − μ)2 σ 2 = E(X 2) − μ2 = 1k (∑ki=1 x2i ) − μ2

X p < 1 fX (x) = 0

X

fX (x) = p x(1 − p)1−x , x = 0, 1

0<

X

(1 − p)

q

Pr(X = 1) = p

Pr(X = 0) = 1 − p = q

μ = E(X) = p σ 2 = V (X) = p(1 − p) = pq

fX (x) p

B(n, p) X fX (x) = (nx)px (1 − p)n−x = (nx) px qn−x

x = 0, 1, ⋯ , n

X

p

n p 00

0

1 α 1 α2

f (X)dx = 1

Pr(X > t) = e−αt Pr(X ≤ t) = 1 − e−αt fX (x) = 1μe −x/μ

x>0

E(X) = μ

σ2 =μ2

Pr(X > s + t∣X > s) = Pr(X > t)

N (μ, σ2 ) μ) 1 exp(− (x− 2σ2 2π σ

2

−∞ < x < ∞ −∞ < μ < ∞ σ>0

X ), −∞ < x < ∞

fX (x) =

x=μ 1 2π σ

x=μ 1 μ σ X V (Z) = σ2Z = 1

σ2 σ

2

N (μ, σ )

Z=

(X−μ) σ

Z

N (0, 1)

E(Z) = 0

x 1 < X < x2 = (x1 − μ)/σ < Z < (x2 − μ)/σ Φ(z)

z z

Φ(z)

Z 1 − Φ(z )

Φ(z) = Pr(Z ≤ z ) ST2334 Notes by Hanming Zhu

1 − Φ(z) = Pr(Z > z ) 100α

∞

α = Pr(Z ≥ zα ) = ∫zα

zα

1 2π

2

exp(−z2 )dx

Pr(Z ≥ zα ) = Pr(Z ≤ −zα ) = α

n→∞ p → 1/2 np > 5 n(1 − p) > 5 σ2 = np(1 − p)

μ = np

X

n→∞ Z=

X−np npq

N (0, 1)

Y N (μ, σ2 ) Pr(X = k) ≈ Pr(k − 12 < X < k + 12 ) Pr(a ≤ X ≤ b) ≈ Pr(a − 12 < X < b + 12) Pr(a < X ≤ b) ≈ Pr(a + 12 < X < b + 12) Pr(a ≤ X < b) ≈ Pr(a − 12 < X < b − 12) Pr(a < X < b) ≈ Pr(a + 12 < X < b − 12) Pr(X ≤ c) = Pr(0 ≤ X ≤ c) ≈ Pr(− 12 < X < c + 12 ) Pr(X > c) = Pr(c < X ≤ n) ≈ Pr(c +

1 2

< X < n + 12 )

n

n

f X (x)

X X

(X 1, X 2, ⋯ , X n)

n fX (x)

fX1 ,X2 ,⋯ ,Xn (x1 , x2 , ⋯ , xn ) = fX 1 (x1 )fX2 (x2 ) ⋯ fXn (xn )

(X1 , X2 , ⋯ , Xn )

(Nn )

X1 , X2 , ⋯ , Xn n

n

N 1

(n ) N

Nn

n

N

1 Nn

ST2334 Notes by Hanming Zhu

(X 1 , X 2, ⋯ , Xn )

X=

X1 , X2 , ⋯ , Xn

n

1 n

n ∑i=1 Xi

x=

X

x1 , x2 , ⋯ , xn

n

n

∑i=1 xi

μ

σ

X

μX = μX σ 2X =

1 n

E( X ) = E(X)

σ2X

V (X ) =

n

V (X) n

X1 , X2 , ⋯ , Xn

n

ϵ ∈ R P(∣ X − μ∣ > ϵ) → 0

n→∞

X Z=

σ2

μ

X −μ σ/ n

σ2 n

μ

n

N (0, 1) X μX = μ

σX =

σ n

Xi , i = 1, 2, ⋯ , n

N (μ, σ2 )

2

N (μ, σn )

X

N (μ, σ2 )

Xi , i = 1, 2, ⋯ , n

n 1 (≥ 30)

n 2

N (μ, σn )

X

n

n 2(≥ 30)

μ1 X1

μ2

σ12

σ22

X2

μX 1 −X 2 = μ1 − μ 2 σ21 n1

σ X 1 −X 2 =

+

σ22 n2

X 1− X 2−(μ 1−μ 2) σ2 1 n1

+

N (0, 1)

σ 2 2 n2

χ2 (n) f Y (y) =

Y

2

χ (n)

n Γ(⋅) 2

Γ(n) =

E(Y ) = n

Y χ (n) 2

n χ (n)

∞ ∫0

n−1 −x

x

e

1 (n/2)−1 −y/2 e 2 n/2 Γ(n/2) y

y>0

Y

Γ(⋅)

n

dx = (n − 1)!

0

n = 1, 2, 3, ⋯

V (Y ) = 2n

N (n, 2n)

Y1 , Y2 , ⋯ , Yk n1 + n 2 + ⋯ + nk

n1 , n2 , ⋯ , nk k ∑i=1 Yi

Y 1 + Y2 + ⋯ + Yk χ

2

k ( ∑i=1 n i)

ST2334 Notes by Hanming Zhu

X2

X N (0, 1) X N (μ, σ 2)

χ2 (1) [(X − μ)/σ ]2

χ2 (1)

X 1, X 2, ⋯ , X n

2

n

Y = ∑i=1 (Xiσ−2 μ)

σ2

μ

Y

χ 2 (n)

χ2 ∞

Pr(Y ≥ c) = ∫c fY (y)dy = α

c Pr(Y ≥ χ2 (n; α))

= ∫χ∞ 2 (n;α) f Y (y )dy

Y

χ 2 (n)

χ2 (n; α)

c

=α

χ2 (n; 1 − α) χ2 (n;1−α)

Pr(Y ≤ χ2 (n; 1 − α)) = ∫0 χ2 (10; 0.9)

fY (y )dy = α

Pr(Y ≥ χ2 (10; 0.9)) = 0.9

Pr(Y ≤ χ2 (10; 0.9)) = 0.1 χ2 (10; 0.9) = 4.865

χ2

(n − 1)S 2 /σ2 S2 =

S

1 n−1

n (Xi − X )2 ∑i=1

(n−1)S 2 σ2

Xi

2

N (μ, σ2 )

i (n−1)S 2 σ2

σ2

n (n−1)S 2 σ2

n−1

χ2 (n − 1)

t Z N (0, 1) Z U /n

U

χ2 (n)

Z

T=

U

Z U /n

T

t

n

t(n)

Γ( n+1 ) t2 n+1 2 ) 2 , −∞ < t < ∞ (1 + nπΓ( n2 ) n

f T (t) = t t

n→

n

2 lim n→∞ f T (t) = 12π e−t /2 ∞ Pr(T ≥ t) = ∫t fT (x)dx

∞

n

Pr(T ≥ t10;0.05 ) = 0.05

t

t10;0.05 = 1.812 T

E(T ) = 0

t(n)

V (T ) =

n n−2

n>2

t Z= Z

T

U=

(n−1)S 2 σ2

χ2 (n −1)

X

S2

t

( X − μ)/(σ/ n ) X −μ = = S/ n (n−1)S 2 /(n − 1) 2 σ

Z U /(n − 1)

n−1

F (n1 , n2 )

F U F

N (0, 1)

U

T =

tn−1

( X −μ) σ/ n

χ 2(n 1)

V

χ2 (n2 )

F =

U /n1 V /n2

(n1 , n2 ) F n /2 n /2

f F (x) =

2 n 1 1 n2 2 Γ( n1 +n ) x(n1 /2)−1 2 n1 n2 (n1 x + n2 )(n 1+n 2)/2 Γ( 2 )Γ( 2 )

ST2334 Notes by Hanming Zhu

x>0

0

E(X) = n 2/(n 2 − 2) V (X) = F

n2 > 2

2n22 (n1 +n2 −2) n 1(n 2−2) 2(n 2 −4)

F (n, m)

n2 > 4

1/F

F (m, n)

F F (n 1, n 2; α)

F F (5, 4; 0.05) = 6.26

Pr(F > 6.26) = 0.05

Pr(F > F (n1 , n2 ; α)) =α

F (5, 4)

F

F (n1 , n2 ; 1 − α) = 1/F (n2 , n1 ; α)

fX (x; θ)

X θ x1 , x2 , ⋯ , xn

X1 , X2 , ⋯ , Xn

f X (x; θ)

x 1 , x 2 , ⋯ , xn

θ

n X = 1n ∑i=1 Xi

max(X1 , X2 , ⋯ , Xn ) W =

1 n

n ∑i=1 (Xi − μ)2

W

X(n) =

μ

Θ = Θ(X 1, X 2, ⋯ , X n)

θ

μ X

μ

X

Θ

x

μ

θ E( Θ) = θ

ΘL

ΘU

ΘL < Θ U

(Θ L, ΘU )

Θ L = X − 2 σn

ΘU = X + 2

θ σ2

σ n

(X − 2 σn ,X +2

θ

θL < θ < θU

θ

θU

θL

σ ) n

μ

θU

Θ Θ θL 1−α

( ΘL , Θ U )

θ Pr( Θ L < θ < ΘU ) = 1 − α

θL < θ < θU

(1 − α)100%

n (1 − α)100%

(1 − α)100% θ

θ

θ (1 − α)

θ

ST2334 Notes by Hanming Zhu

(≥ 30)

n

X Z=

X −μ σ/ n

N (μ,

σ2 ) n

N (0, 1)

X −μ < zα/2 ) = 1 − α Pr(−zα/2 < σ/ n

Pr(X − zα/2 ( σ n ) < μ < X + zα/2 (

σ )) n

=1 − α

z0.025 = 1.96

X

σ2

n X − z α/2(

(1 − α)100%

μ

σ σ ) ) < μ < X + z α/2( n n

μ ∣ X − μ∣

Pr(∣ X − μ∣ < zα/2

σ n

) =1 − α 1−α

e

X −μ S/ n

σ ) n

n ≥ (zα/2 σe )2

e

T =

e ≥ zα/2 (

S2

T

tn−1

Pr(−tn−1;α/2 < T < tn−1;α/2 ) = 1 − α Pr(−tn−1;α/2 <

( X −μ) S/ n

Pr(−tn−1;α/2 Sn

< X − μ < tn−1;α/2

Pr(X − tn−1;α/2

X

S n

< tn−1;α/2 ) = 1 − α S ) n

< μ < X + tn−1;α/2

=1−α S n

) =1 − α

n < 30

S σ2

(1 − α)100%

μ X − t n−1;α/2(

n > 30

N (0, 1)

t

X − z α/2(

μ1

σ 21

S S ) < μ < X + tn−1;α/2 ( ) n n

μ2

σ12

n

(1 − α)100%

μ

S S ) ) < μ < X + z α/2( n n

σ22

X1 − X2

μ1 − μ2

σ22 n 1 ≥ 30, n2 ≥ 30 σ2

σ2

(X 1 − X 2 ) N (μ1 − μ2 , n11 + n22 )

ST2334 Notes by Hanming Zhu

Pr(−zα/2 <

( X 1 − X 2) − (μ 1 − μ 2) σ12 n1

(1 − α)100%

< zα/2 ) = 1 − α

μ1 − μ 2

(X 1 − X 2 ) − zα/2

σ 21

σ2

+ n 22

σ2 σ21 + 2 < μ1 − μ 2 < (X 1 − X 2 ) + zα/2 n2 n1

σ2 σ21 + 2 n2 n1

σ22

n 1 ≥ 30, n 2 ≥ 30 σ 21

σ22

S12

S 22

(X 1 − X 2 ) − zα/2

σ 21

(1 − α)100%

μ1 − μ2

S2 S12 + 2 < μ1 − μ 2 < (X 1 − X 2 ) + zα/2 n2 n1

S2 S12 + 2 n2 n1

σ22 n 1 ≤ 30, n2 ≤ 30 σ 21

=

σ22

=σ

2

(X 1 − X 2 ) N (μ1 − μ2 , σ2 ( n11 +

1 )) n2

Z=

( X 1 − X 2) − (μ 1 − μ 2) σ 2 ( n11 +

1 n2

)

σ2 Sp2 =

(n1 −1)S21 +(n2 −1)S22 σ2

χ n21 +n2 −2 S2p

T =

( X 1− X 2)−(μ 1−μ 2)

σ2

tn 1+n 2−2

Sp2 (n1 +n1 ) 1

(n 1 − 1)S12 + (n 2 − 1)S22 n1 + n2 − 2

2

t

Pr(−tn1 +n2 −2;α/2 <

( X 1 − X 2) − (μ 1 − μ 2) S 2p ( n1 + 1

(1 − α)100%

1 ) n2

< tn 1+n 2−2;α/2 ) = 1 − α

μ1 − μ2

(X 1 − X 2 ) − tn 1+n 2−2;α/2 Sp

tn 1+n 2−2;α/2

zα/2

(X 1 − X 2 ) − z α/2 S p

1 1 < μ1 − μ 2 < (X 1 − X 2 ) + tn 1+n 2−2;α...