Probability Theory Formula Sheet PDF

Probability Theory. P( A B) = P ( A) + P ( B) P( A B ), P( A B ) = P ( A B )P ( B ) = P ( B A )P ( A ) The Total Probability Rule...

Probability Theory. P( A B) = P ( A) + P ( B) P( A B ), P( A B ) = P ( A B )P ( B ) = P ( B A )P ( A ) The Total Probability Rule Bayes’ Rule P ( A E j )P ( E j ) P ( A) = P ( A E1 ) + P ( A E2 ) + + P( A En ) P ( Ej A) = n P ( A Ei )P ( Ei ) i =1 E ( X ) = E ( X E)P ( E) + E ( X EC )P ( EC ) = E ( X E1 )P ( E1 ) + E ( X E2 )P ( E2 ) + + E ( X En )P ( En ) n n E(X) = Xi P ( Xi ) E (Y ) = E ( g( X )) = g ( Xi ) P ( Xi ) i =1 i =1 ⇣ ⌘ n Var ( X ) = E ( X E ( X ))2 = ( Xi E ( X ))2 P ( Xi ) Cov( X, Y ) = sX,Y = E ([ X E ( X )][Y E (Y )]) i =1 Cov( X,Y ) s r( X, Y ) = r X,Y = = sXX,Y sY s 2 ( X ) s 2 (Y ) E ( aX + bY ) = aE ( X ) + bE (Y ), E ( aX + c) = aE ( X ) + c, s2 ( X ) = E ( X 2 ) (E ( X ))2 s2 ( aX + bY ) = a2 s2 ( X ) + b2 s2 (Y ) + 2ab Cov( X, Y ) s2 ( aX + c) = a2 s2 ( X ) If X1 , X2 ,..., Xn are independent, Cov( X, Y ) = E ( XY ) E ( X ) E (Y ) ✓ n ◆ n s 2 Xi = s 2 ( Xi ) i =1 i =1 X B(n, p) X Poisson(l) n! lk p( x ) = P ( X = x ) = (n x )!x! p x (1 p)n x P( X = k) = k! e l, k = 0, 1, 2,... a+b (b a )2 X U [ a, b],E ( X ) = 2 , Var ( X ) = 12 0, for x a 1 b a for a x b f (x) = F(x) = x a b a, for a x b 0 otherwise 1, for x b. X N (µ, s2 ) Y = eX LN (µ, s2 ) ⇣ 2 ⌘ 1 ( x µ) 1 2 f (x) = 2ps exp 2s2 , for x E (Y ) = E e X = eE(X )+ 2 Var(X ) = eµ+0.5s c2 distribution: If z1 , z2 ,.., zn are standard normal independent random variables, then kn = z21 + z22 + + z2n is c2 distributed with n degrees of freedom. kn /n F-distribution: If kn c2 (n) and km c2 (m) are independent, then km /m F (n, m). z t-distribution: If z is standard normal, kn c2 (n), z and kn are independent then kn /n has t-distribution with n degrees of freedom. n n 2 1 s ( Xi X) X= n Xi sX = s ,s n X = n , s2 = i =1 n 1 i =1 nl L 1 Nl X l Stratified Sampling: X l = nl Xil Xs = N i =1 l =1 L ⇣ ⌘2 nl Nl 1 2 1 2 s2X = N nl s l , where s2l = nl 1 Xil Xl s l =1 i =1 n n 3 1 n i =1 ( Xi X) s XY = n 1 ( Xi X )(Yi Y) Skew = (n 1)(n 2) s3 i =1 n 4 n ( n +1) i =1 ( Xi X) 3( n 1)2 Kurt E = (n 1)(n 2)(n 3) s4 (n 2)(n 3) Hypothesis Testing. ( X1 X2 ) ( µ 1 µ 2 ) (n1 1)s21 +(n2 1)s22 Equal but unknown variances t= ✓ 2 ◆ , d f = n1 + n2 2, s2p = n1 + n2 2 sp s2p n +n 1 2 2 ( X1 X2 ) ( µ 1 µ 2 ) (s21 /n1 +s22 /n2 ) Unequal and unknown variances t= ✓ 2 2 ◆ , df = s41 /n31 +s42 /n32 s 1 + s2 n 1 n 2 ( n 1) s2 Test of a single variance TS = s02 c2n 1 s21 Test to compare two variances TS = s22 F ( d f 1 , d f 2 ), d f i = n i 1, rs n 2 Correlation test TS = tn 2 1 rs2 Regression Analysis. Univariate Regression Multivariate Regression N i =1 ( yi y )( xi x) s xy 1X b̂ 1 = N 2 = s2x b̂ 0 = y b̂ 1 x b̂ = ( X X ) Y i =1 ( x i x ) 1 1 xi2 ⇥ ⇤ SE b̂ 1 = s ( x i x )2 SE b̂ 0 = s N ( x i x )2 Var b̂ = E ( b̂ b)( b̂ b) = (X X) 1X E [uu ] X ( X X ) 1 = (X X) 1X ( s2 I ) X ( X X ) 1 = s2 ( X X ) 1 N 1 û û s2 = N 2 û2i s2 = N k 1 i =1 h i ESS RSS (yi ŷi )2 2 n 1 LLF R2 = TSS =1 TSS =1 ( y i y )2 , R =1 n k 1 1 R2 , pseudo R2 = 1 LLF0 RRSS URSS n k 1 F-test TS = URSS m F (m, n k 1) 2 2 TS = R1u RR2r n mk 1 F (m, n k 1) u ( R b̂ q) ( R( X X ) 1 R ) 1 ( R b̂ q) Wald-test ms2 F (m, n k 1) n n Durbin-Watson test DW = (ûi ûi 1 )2 / û2i i =2 i =1 White test nR 2 cm 2 Breusch-Godfrey test (N r ) R2 c2r h i 2 (kurt 3)2 Jaque-Berra test W = n skew6 + 24 c22 RSS ( RSS1 + RSS2 ) n 2(k +1) Chow test TS = RSS1 + RSS2 k +1 F (k + 1, n 2( k + 1) Logit model Pi = F (zi ), zi = b 0 + b 1 X1i + + b k Xki + ui , F ( zi ) = 1 1+exp( zi ) , XPij = F (z)(1 F (z)) b j zi 2 Probit model Pi = F(zi ), zi = b 0 + b 1 X1i + + b k Xki + ui , F ( zi ) = 12p e z /2 dz, XPij = f(z) b j