Formula Sheet. December 8, 2011
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1 Formula Sheet December 8, 2011 Abtract I type thi for your coveice. There may be error. Ue at your ow rik. It i your repoible to check it i correct or ot before uig it. 1 Decriptive Statitic 1.1 Cetral Tedecy 1. (Arithmetic) Mea: P N Populatio: = x i P N Sample: x = x i 2. Weighted mea:x w = P w ix i P w i 3. Media: The 0:5 ( + 1)th data. If i eve, the we take 0:5 ( + 1)th data directly If i odd, the we take mea of 0:5th ad (0:5) + 1th data 4. Mode: The mot frequet data. (However, o mode if every data oly appear oce.) 1.2 Diperio 1. Rage: Rage = X Larget X Smallet 2. Iterquartile Rage: IQR = Q 3 Q 1 where Q 3 = 0:75( + 1)th data ad Q 1 = 0:25( + 1)th data. If + 1 i multiple of 4, the we take 0:75( + 1)th data ad the 0:25( + 1)th data a Q 3 ad Q 1 : If + 1 i ot multiple of 4, the we have to iterpolate. 3. Variace: P N Populatio: 2 = (X i ) 2 N 1
2 P Sample: 2 = (x i x) Stadard Deviatio rp N Populatio: = (x i ) 2 N P Sample: = (x i x) Coe ciet of variatio: CV = X 100% 1.3 Relatiohip betwee two variable 1. Covariace: P N Populatio: XY = (x i X ) (y i Y ) P N Sample: XY = (x i x) (y i y) 1 2. Correlatio coe ciet Populatio: XY = XY X Y Sample: r XY = XY X Y Note: 1 XY 1 ad 1 r XY 1 2 Probability Axiom of probability: 1. For ay evet E, 0 P (E) 1: 2. For ay evet E coitig outcome O 1 ; : : : ; O K, P (E) = P K P (O K) : 3. For ample pace S, P (S) = 1. Mutually excluive evet: A ad B are mutually excluive if ad oly if P (A \ B) = 0 Collectively exhautive evet: A 1, A 2,... ; A K are collectively exhautive if Complemet rule: Odd ratio i favor of A i Additio rule: P (A 1 [ A 2 [ : : : [ A K ) = 1 P (A) = 1 P A ; P A = 1 P (A) Odd (A) = P (A) P A = P (A) 1 P (A) P (A \ B) = P (A) + P (B) P (A [ B) P (A [ B) = P (A) + P (B) P (A \ B) 2
3 Coditioal probability: P (AjB) = P (A \ B) ; P (BjA) = P (B) P (A \ B) P (A) Multiplicatio rule: P (A \ B) = P (AjB) P (B) = P (BjA) P (A) Idepedece: Evet A ad B are idepedet if ad oly if P (A \ B) = P (A) P (B) Note: Thi implie that evet A ad B are idepedet if P (AjB) = P (A) P (BjA) = P (B) Law of total probability: If E 1, E 2,... ; E K are mutually excluive ad collectively exhautive, the P (A) = P (A \ E 1 ) + P (A \ E 2 ) + + P (A \ E K ) Baye theorem: = P (AjE 1 ) P (E 1 ) + P (AjE 2 ) P (E 2 ) + + P (AjE K ) P (E K ) P (BjA) = P (E i ja) = P (AjB) P (B) P (A) P (AjE i ) P (E i ) P (AjE 1 ) P (E 1 ) + P (AjE 2 ) P (E 2 ) + + P (AjE K ) P (E K ) 3 Dicrete Radom Variable For a radom variable X, probability ditributio fuctio i p (x) = P (X = x) ad the cumulative probability fuctio: F (x) = P (X x) = X p (x) ax Expectatio value Variace: E (X) = X xp (x) V ar (X) = X [x E (X)] 2 p (x) 3
4 Propertie of expectatio ad variace fuctio: E (g (X)) = X g (x) p (x) E (a + bx) = a + bex V ar (a + bx) = b 2 V arx Joit probability fuctio for radom variable X ad Y Covariace Correlatio p (x; y) = P (X = x; Y = y) Cov (X; Y ) = E [(X E (X)) (Y E (Y ))] = X X [x E (X)] [y E (Y )] p (x; y) x y = Corr (X; Y ) = Propertie of expectatio ad variace fuctio: Cov (X; Y ) p V arx p V ary E (ax + by ) = ae (X) + be (Y ) V ar (ax + by ) = a 2 V arx + b 2 V ary + 2abcov (X; Y ) 4 Cotiuou Radom Variable For a radom variable X, the cumulative probability fuctio: F (x) = P (X x) = X ax p (x) Hece, P (a < X < b) = F (b) F (a) Propertie of expectatio ad variace fuctio for radom variable X ad Y E (a + bx) = a + bex V ar (a + bx) = b 2 V arx E (ax + by ) = ae (X) + be (Y ) V ar (ax + by ) = a 2 V arx + b 2 V ary + 2abcov (X; Y ) If X i ormally ditributed with mea ad variace 2, we deote X N ; 2 Normal ditributio i ymmetric about mea, bell-haped, mea=media=mode. Theorem: If X N (; 2 ), the Z = X N (0; 1) 4
5 5 Samplig Let X 1 ; X 2 ; :::; X be radom ample from a populatio with mea ad variace 2. Sample mea X = 1 X X i ad tadard error of the mea Fact: If populatio i ormal, the X = X N ; 2 Fact: If poulatio i NOT ormal, the by cetral limit theorem, whe ample ize i large, the X N ; 2 6 Etimatio Coider a populatio parameter ad it poit etimator ^. A etimator ^ i ubiaed if E ^ = : Fact: ample mea X i a ubiaed etimator of populatio mea. Fact: ample variace 2 i a ubiaed etimator of populatio variace 2. For two ubiaed etimator ^ 1 ad ^ 2, etimator ^ 1 i more e ciet if V ar V ar ^2 Iterval etimator: 100(1 ) % co dece iterval etimator with kow 2 i ^1 X Z =2 p where ME = Z =2 p, UCL = X + Z =2 p ad LCL = X Z =2 p 100(1 ) % co dece iterval etimator with ukow 2 uder ample ize ad i X t 1;=2 where ME = t 1;=2, UCL = X + t 1;=2 ad LCL = X t 1;=2 5
6 7 Hypothei Tetig Give populatio variace 2, hypothei tetig with igi cace level with radom ample of ize ad ample mea x. Kow 2 Two-tail Tet Upper Tail Lower Tail Null Hypothei H 0 = 0 = 0 or 0 = 0 or 0 Alterative Hypothei H 1 6= 0 > 0 < 0 X critical value 0 z =2 p 0 + z p 0 z p Deciio: Reject if x > 0 + z =2 p or x < 0 z =2 p x > 0 + z p x < 0 z p tet tatitic z = x 0 = z = x 0 = z = x 0 = Z critical value z =2 +z z Deciio: Reject if z > z =2 or z < z =2 z > z z < z p-value 2P (Z > jzj) P (Z > z) P (Z < z) Deciio: Reject if > p Without kowig populatio variace 2, hypothei tetig with igi cace level with radom ample of ize, ample mea x ad ample tadard deviatio 2 : Ukow 2 Two-tail Tet Upper Tail Lower Tail Null Hypothei H 0 = 0 = 0 or 0 = 0 or 0 Alterative Hypothei H 1 6= 0 > 0 < 0 X critical value 0 t 1;=2 0 + t 1; 0 t 1; Deciio: Reject if x > 0 + t 1;=2 x > 0 + t 1; p x < or x < 0 t 1;=2 0 t 1; t-tet tatitic t = x 0 = t = x 0 = t = x 0 = T critical value t 1;=2 +t 1; t 1; Deciio: Reject if t > t 1;=2 or t < t 1;=2 t > t 1; t < t 1; p-value 2P (T > jtj) P (T > t) P (T < t) Deciio: Reject if > p 6
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