Math Tricks. Basic Probability. x k. (Combination - number of ways to group r of n objects, order not important) (a is constant, 0 < r < 1)

Transcription

1 Math Trcks r! Combato - umbr o was to group r o objcts, ordr ot mportat r! r! ar 0 a r a s costat, 0 < r < k k! k 0 EX E[XX-] + EX Basc Probablt 0 or d Pr[X > ] - Pr[X ] Pr[ X ] Pr[X ] - Pr[X ] Proprts o EX EX or d EaX + b aex + b EX + X X EX + EX EX I a such that Pr[X a], th EX > a I b such that Pr[X b], th EX < b Proprts o VarX VarX E[ - ] EX - [EX] VarX or VarX 0 VaraX + b a VarX d Pr[X ].., X ol taks o o valu VarX + X X VarX + VarX VarX all X ar dpdt o 8

2 Skwss - E[ - 3 ] Skwss Coct - dvd b 3 Kurtoss - E[ - 4 ] Dgr o Ecss Kurtoss Kurtoss o ormal dstrbuto s 3 Ctral Momt - r E[ - r ], r Raw Momt - E[ r ] Fuctos o Radom Varabls r s a ucto o radom varabl r must b mootoc... crasg or dcrasg Ivrs ucto s s Dscrt pd: g Pr[Y ] Pr[r ] : r d Cotuous pd: g G d Cotuous cd: G Pr[Y ] Pr[r df s ds ds s ds d d : r d Pr[X s] Fs Multpl Varabls Jot Dstrbuto -, PrX ad Y, 0, b d Pr[a b ad c d] a c, dd Jot Cumulatv Dstrbuto - F, PrX ad Y, X Y, dd Margal Dstrbuto -, o 8

3 Covarac - masurs lar rlatoshp btw ad CovX,Y E[ - - ] E - X ad Y dpdt, dd CovX,Y 0 dos't work th othr wa aroud VarX + Y VarX + VarY + CovX,Y VaraX + by + c a VarX + b VarY + abcovx,y Corrlato - ρ ρ ρ - Cov X, Y Var X + Var Y prct lar rlatoshp btw X ad Y prct vrs lar rlatoshp btw X ad Y.., slop s gatv Idpdc - PrA & B PrA B PrA PrB CovX,Y 0, Codtoal Probablts - PrX Y Pr X & Y Pr Y \ 3 4 Margal Margal , Pr[X Y ] 0./ Pr[X Y ] 0.3/ Pr[A B] Pr[B A] Pr[A] Multpl Codtos - Pr[A B C] Pr[A B] Pr[C] I A & B ar dpdt... Pr[A C] Pr[B C] Pr[C] Codtoal Epctato - EX Y d or.g., EX Y Law o Itratd Epctatos - a lvl o Y ad calculat EX Y, th tak pctd valu o ths to d EX; us ou hav jot dstrbuto ad ca't gt X b tsl EX E [EX Y] E X Y d Codtoal Varac - VarX Y E[X - EX Y ] 3 o 8

4 [ E X Y ] d or E X Y ] [ Multvarat Dstrbutos E E E E T Cov E [ ] Smmtrc matr - j j Cov, j j or Var j Idpdt - Σ s dagoal matr wth varacs ol j Dscrt Dstrbutos Broull - smpl s/o or succss/alur; Pr[succss] p; {0, } pd: p p EX p VarX p p Bomal - succsss trals; Pr[succss] p; sum o d Broull s; {0,,, } pd: p p EX p VarX p p D Movr-Laplac Ctral Lmt Thorm - X ~ Bomal,p, th or d a b, p Pra b Pr[ p + a pq p + b pq ] Φ b Φ a as pq Eampl: B00,0.05, Pr[ 0] Pr[0 0] 0 p p Fd a ad b pq pq Pr[0 0] Φ Φ ot: Sam calculato usg Bomal lds Gomtrc - trals utl rst succss; Pr[succss] p; {,, 3, } 4 o 8

5 pd: p p EX p VarX p p gatv Bomal - trals utl r th succss; Pr[succss] p; {r, r+, } pd: p r r r p EX r p VarX r p p Posso - succsss cotuous ara, volum, tm, tc.; {0,,, } pd:! EX VarX Hprgomtrc - bomal wth dpdt trals; m # o possbl succsss; # o dsrd succsss; total # o trals possbl; # o trals usd pd: m m EX m VarX m m Cotuous Dstrbutos Uorm - qual probablt o occurrc awhr rag [a,b] 0 b a, a b, ls EX a + b Epotal - tm utl rst occurrc; closl rlatd to Posso, 0 0, ls EX Gamma Erlag - tm utl k th occurrc; sum o k d Epotal k α α k Γ Γ α β k / β VarX b a VarX EX k αβ VarX k αβ α OTE: Γ α d α Γ α α! α s tgr ormal - smmtrc about ma 0 EX VarX π 5 o 8

6 6 o 8 Bvarat ormal - + p, ρ ρ ρ π, Margals ar also ormal: ~, Codtoal Dstrbuto o X Y also ormal: ~ +, ρ ρ Multvarat ormal - p T π Spcal Cass: dpdt: p π d, : p π Proprts: Y A + b ~ A + b, AΣA T Partto to substs: Σ Σ T Margal dstrbutos also ormal: ~,Σ ad ~,Σ ot: Los o Σ ulss th'r & ar dpdt Codtoal dstrbuto: ~, + m m m m m m m m m

7 Spcal Dstrbutos Stadard ormal - ormal wth ma 0 ad varac π EX 0 VarX X ~, z ~ 0, φz cd o stadard ormal Ch Squard - squard stadard ormal s ch squard wth dgr o rdom z ~ 0, X z ~ χ d Γ, > 0,,,... EX VarX X ~ χ m ad X ~ χ m ; X ad X ar dpdt, th X + X ~ χ m+ F - rato o ch squard dstrbutos dvdd b thr dgrs o rdom X ~ χ ad X ~ χ F m X X / m ~ F m, d / m / k, > 0, m,,,... EX VarX m+ / m + t - lattr tha a stadard ormal wdr tals z ~ 0, ad X ~ χ t + z X / ~ t k +,,,... EX VarX 7 o 8

8 Brthda Problm Pr popl group o popl hav b-da o sam da - PrEvro's b-das ar drt ! Pr w/ sam b-da o 8