STA 4032 Final Exam Formula Sheet
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1 Chapter 2. Probability STA 4032 Fial Eam Formula Sheet Some Baic Probability Formula: (1) P (A B) = P (A) + P (B) P (A B). (2) P (A ) = 1 P (A) ( A i the complemet of A). (3) If S i a fiite ample pace with equal probability for each outcome, the for ay evet A, we have Coditioal Probability: Idepedece: P (A) = N(A) N(S) = P (A B) = umber of outcome i A total umber of outcome i S P (A B), P (B) > 0. P (B) Two evet A ad B are idepedet if P (A B) = P (A)P (B). If evet A 1, A 2,, A are idepedet, the Chapter 3. Dicrete Ditributio P ( i=1a i ) = P (A 1 )P (A 2 ) P (A ). Probability Ma Fuctio : f() = P (X = ), { 1, 2, }, which i the et of all poible value for X. Ditributio Fuctio : F () = P (X ). Epected Value ad Variace (X i a dicrete radom variable): (1) µ = E(X) = P (X = ) = f(), E[h(X)] = h()f() (2) 2 = V (X) = E(X 2 ) (E(X)) 2, = 2 = V (X) Special Dicrete Radom Variable Biomial : f() = P (X = ) = ( ) p (1 p), = 0, 1,,, E(X) = p, V (X) = p(1 p) Geometric : f() = p(1 p) 1, = 1, 2,, E(X) = 1 p, V (X) = 1 p p 2 Chapter 4. Cotiuou Radom Variable ad Probability Ditributio 1
2 A radom variable X i called cotiuou if there i a oegative fuctio f, called the probability deity fuctio of X, uch that (1) f() 0 (2) f()d = 1 (3) P (a < X b) = b a f()d. The cumulative ditributio fuctio of X i defied by F () = P (X ) = We have the formula P (a < X b) = F (b) F (a). f(u)du. Epectatio ad Variace of Cotiuou Radom Variable The mea or epected value of a cotiuou radom variable X i defied by The variace of X i defied by ad the tadard deviatio of X i We have the followig formula: (1) 2 = V (X) = E(X 2 ) µ 2. (2) For ay fuctio h, µ = E(X) = 2 = V (X) = E[(X µ) 2 ] = E[h(X)] = Special Cotiuou Radom Variable = V (X) f()d. h()f()d. ( µ) 2 f()d (i) A radom variable X i aid to be ormal with parameter µ ad 2 (deoted by N(µ, 2 ) ) if it probability deity fuctio i give by f() = 1 e ( µ)2 2 2, < < 2π We have E(X) = µ ad V (X) = 2. If X ha a ormal ditributio N(µ, 2 ), the Z = (X µ)/ i a tadard ormal radom variable N(0, 1) with mea 0 ad variace 1. The ditributio fuctio of X ca be epreed by ( X µ F X (a) = P (X a) = P a µ ) = P ( Z a µ ) ( a µ = Φ where Φ i the ditributio fuctio of Z. We alo have ( ) b µ P (a X b) = F (b) F (a) = Φ Φ 2 ). ( a µ ).
3 (ii) A radom variable X i aid to be epoetial with parameter λ if it deity fuctio i give by { λe f() = λ, 0 0, otherwie. The ditributio fuctio F () of X i give by { 1 e F () = P (X ) = λ, 0 0, otherwie. The epected value ad variace of X are E(X) = 1 λ, V (X) = 1 λ 2. It atifie the memoryle property, for poitive ad t, P (X > + t X > t) = P (X > ) Liear Fuctio of Radom Variable Give radom variable X 1, X 2,..., X p ad cotat c 1, c 2,..., c p, i a liear combiatio of X 1, X 2,..., X p. Y = c 1 X 1 + c 2 X c p X p Some related formula: (1) E(Y ) = c 1 E(X 1 ) + c 2 E(X 2 ) + + c p E(X p ). (2) If X 1, X 2,..., X p are idepedet, the V (Y ) = c 2 1V (X 1 ) + c 2 2V (X 2 ) + + c 2 pv (X p ). (3) If X 1, X 2,..., X p are idepedet, ormal radom variable with E(X i ) = µ i ad V (X i ) = 2 i for i = 1, 2,..., p, the Y = c 1 X 1 + c 2 X c p X p i a ormal radom variable with ad E(Y ) = c 1 µ 1 + c 2 µ c p µ p V (Y ) = c c c 2 p 2 p. Chapter 7. Poit Etimatio of Parameter ad Samplig Ditributio If X 1,, X i a radom ample from a populatio with ormal ditributio N(µ, 2 ), the the ample mea X = X1+X2+ +X ha a ormal ditributio N(µ, 2 ) with mea µ X = µ ad tadard deviatio X = /. We ca tadardize X to get Z = X µ / N(0, 1). Cetral Limit Theorem: If X 1,, X i a radom ample of ize take from a populatio (either fiite or ifiite) with mea µ ad fiite variace 2, ad if X i the ample mea, the limitig ditributio of Z = X µ / 3
4 i the tadard ormal ditributio a. Ubiaed Etimator: The poit etimator ˆΘ i a ubiaed etimator for the parameter θ if E( ˆΘ) = θ. If the etimator i ot ubiaed, the the differece E( ˆΘ) θ i called the bia of the etimator ˆΘ. The mea quared error of a etimator ˆΘ of the parameter θ i defied a MSE( ˆΘ) = E( ˆΘ θ) 2 = V ( ˆΘ) + (bia) 2, where V ( ˆΘ) i the variace of ˆΘ. ˆΘ 1, ˆΘ 2, the relative efficiecy of ˆΘ 2 to ˆΘ 1 i defied a For two etimator MSE( ˆΘ 1 ) MSE( ˆΘ 2 ). If thi relative efficiecy i le tha 1, the ˆΘ 1 i a more efficiet (better) etimator of θ tha ˆΘ 2. Method of Momet: Let X 1, X 2,, X be a radom ample from the propbability ditributio f() (pdf or pmf). The kth populatio momet i E(X k ), k = 1, 2,.... The correpodig kth ample momet i 1 Xi k, k = 1, 2,. i=1 Let the populatio ditributio have m ukow parameter θ 1, θ 2,..., θ m. the momet etimator ˆΘ 1, ˆΘ 2,..., ˆΘ m are foud by equatig the firt m populatio momet to the firt m ample momet ad olvig the reultig equatio for the ukow parameter. Maimum Likelihood Etimator: Suppoe that X i a radom variable with probability ditributio f(; θ), where θ i a igle ukow parameter. Let 1, 2,, be the oberved value i a radom ample of ize. The the likelihood fuctio of the ample i L(θ) = f( 1 ; θ) f( 2 ; θ) f( ; θ) The maimum likelihood etimator (MLE) of θ i the value of θ that maimize the likelihood fuctio L(θ).Baically, we olve the equatio ad it olutio give the MLE of θ. d l L(θ) dθ = 0 Chapter 8. Statitical Iterval for a Sigle Sample Cofidece iterval for µ whe i kow: If i the ample mea of a radom ample of ize from a ormal populatio with kow variace 2, a 100(1 α)% CI o µ i give by z α/2 µ + z α/2, where z α/2 i the upper 100α/2 percetage poit of the tadard ormal ditributio. 4
5 If i ued a a etimate of µ, we ca be 100(1 α)% cofidet that the error µ will ot eceed a pecified amout E whe the ample ize i A 100(1 α)% upper-cofidece boud for µ i ( zα/2 = E ) 2 µ u = + z α ad a 100(1 α)% lower-cofidece boud for µ i z α = l µ Cofidece iterval o µ whe 2 i ukow: Let X 1,, X be a radom ample from a ormal ditributio N(µ, 2 ) with ukow µ ad 2. The radom variable T = X µ S/ ha a t ditributio with 1 degree of freedom. A 100(1 α)% CI o µ (with ukow 2 ) i give by t α/2, 1 µ + t α/2, 1, where t α/2, 1 i the upper 100α/2 percetage poit of the t ditributio with 1 degree of freedom. A 100(1 α)% upper-cofidece boud for µ i µ u = + t α, 1 ad a 100(1 α)% lower-cofidece boud for µ i t α, 1 = l µ CI for populatio proportio p: If ˆp i the proportio of obervatio i a radom ample of ize that belog to a cla of iteret, a approimate 100(1 α)% cofidece iterval o p i ˆp z α/2 p ˆp + z α/2 where z α/2 i the upper α/2 percetage poit of the tadard ormal ditributio. The Choice of Sample Size: ( zα/2 ) 2 = E The approimate 100(1 α)% lower ad upper cofidece boud are ˆp z α p ad p ˆp + z α 5
6 Chapter 9. Tet of Hypothee for a Sigle Sample (Up to Sectio 9.3) α = P (type I error)= P (reject H 0 whe H 0 i true). β = P (type II error)= P (fail to reject H 0 whe H 0 i fale). Tet o the mea with kow variace: Null Hypothei : H 0 : µ = µ 0 Tet Statitic: Z 0 = X µ 0 / Alterative hypothei Rejectio regio H 1 : µ µ 0 z 0 > z α/2 or z 0 < z α/2 H 1 : µ > µ 0 H 1 : µ < µ 0 z 0 > z α z 0 < z α P-value Method: 2[1 Φ( z 0 )], for a two-tailed tet : H 0 : µ = µ 0 v H 1 : µ µ 0 P = 1 Φ(z 0 ), for a upper-tailed tet : H 0 : µ = µ 0 v H 1 : µ > µ 0 Φ(z 0 ), for a lower-tailed tet : H 0 : µ = µ 0 v H 1 : µ < µ 0 If P α, the H 0 would be rejected. If P > α, we will fail to reject H 0. Probability of Type II Error: where δ = µ µ 0. Choice of Sample Size: β = Φ ( z α/2 δ ) ( Φ z α/2 δ ) (z α/2 + z β ) 2 2 δ 2, δ = µ µ 0 Tet o µ with ukow variace: Null Hypothei : H 0 : µ = µ 0 Tet Statitic: T 0 = X µ 0 S/ Let t 0 be the oberved value of T 0 : t 0 = µ 0 / Alterative hypothei H 1 : µ µ 0 H 1 : µ > µ 0 H 1 : µ < µ 0 Rejectio regio t 0 > t α/2, 1 or t 0 < t α/2, 1 t 0 > t α, 1 t 0 < t α, 1 6
7 P-value: 2P (T 1 > t 0 ), for a two-tailed tet : H 0 : µ = µ 0 v H 1 : µ µ 0 P = P (T 1 > t 0 ), for a upper-tailed tet : H 0 : µ = µ 0 v H 1 : µ > µ 0 P (T 1 < t 0 ), for a lower-tailed tet : H 0 : µ = µ 0 v H 1 : µ < µ 0 Here T 1 i the t ditributio with 1 degree of freedom. Oe ca ue Table V (o page 745) i the Appedi to fid critical value ad the correpodig tail area. If P α, the H 0 would be rejected. If P > α, we will fail to reject H 0. Geeral 7-Step Procedure for Hypothei Tet: 1. From the problem cotet, idetify the parameter of iteret. 2. State the ull hypothei H Specify a appropriate alterative hypothei H Determie a appropriate tet tatitic. 5. State the rejectio criteria for the ull hypothei. 6. Compute ay eceary ample quatitie, ubtitute thee ito the equatio for the tet tatitic, ad compute that vale. 7. Draw cocluio: Decide whether or ot H 0 hould be rejected ad report that i the problem cotet. 7
x z Increasing the size of the sample increases the power (reduces the probability of a Type II error) when the significance level remains fixed.
] z-tet for the mea, μ If the P-value i a mall or maller tha a pecified value, the data are tatitically igificat at igificace level. Sigificace tet for the hypothei H 0: = 0 cocerig the ukow mea of a populatio
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