1.2 The Mean, Variance, and Standard Deviation. x x. standard deviation: σ = σ ; geometric series is. 1 x. 1 n xi. n n n

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1 . The Mea, Varace, ad Sadard Devao mea: µ f ( u f ( u ukf ( uk S k k varace: ( f ( ( u f ( u ( u f ( u f ( S sadard devao: geomerc seres s [emrcal dsrbuo s] samle mea: [emrcal dsrbuo s] varace: [emrcal dsrbuo s] samle varace: S f <, ower: ( + seres:, X v ( ( ( s ( + f ( + ( + 6. Proeres of Probabl P(A-P(A muuall elcusve whe A B em se, so P( A B P( A + P( B P(A+P(A P( A B P( A + P( B P( A B, P( A B P( A + P( A' B, P( A' B P( B P( A B P( A B C P( A + P( B + P( C P( A B P( A C P( B C + P( A B C h ( A P( A m ( S Q: dvde a le segme o wo ars, rob. oe ar s a leas wce as log as aoher, s /. Mehods of Eumerao Ordered w/o Relaceme (ermuao of objecs ake r a a me: ags osos,..4 ou of 9 r Ordered w/ Relaceme: (combos of lock, zza ogs Uordered w/o Relaceme (combao of objecs ake r a a me: games, cards, dff. kds of cad seleced a radom Uordered w/ Relaceme: ( + r! r!(!! Pr (-dff. color flags ou of 4 4 ( r! C r! ( r (world seres r!( r!. Codoal Probabl P( A B of eve A gve B occurred: P( A B, f P( B > P( B robabl ha wo eves boh occur (mullcao rule: P( A B P( A P( B A, P( B A P( B P( A B P( A B C P( A P( B A P( C A B Q: Prob. of drawg wo hears: P( A B P( A P( B A (rob. of oe, mes rob. of drawg secod gve cards lef P(( A B' P( A B rob. of kgs -card had w/ a leas kgs: A or 4 Ks, B,, or 4 Ks, ( A ( ( + ( 4( 9 P( A B ( B ( ( + ( ( + ( 4( dce roll ls all oucomes 6 decks, draw 6 cards, all dffere f P(A rob he mach s -P(A

2 .4 Ideede Eves ff P( A B P( A P( B Also, hese are deede: A ad B, A ad B, A ad B A, B, C are muuall deede ff: arwse deede ( P( A B P( A P( B, P( A C P( A P( C, P( B C P( B P( C ad P( A B C P( A P( B P( C Also, hese are deede: A ad (BC, A ad (BuC, A ad (B C, A ad B ad C mu. de. Q: P( A B P( A + P( B P( A P( B f de. laers kck fooball: rob. eacl oe makes s P( A A' A' + P( A' A A' + P( A' A' A (ad so o co ossed mes: rob. of a sequece s (/^, heads of s ( choose *(/^. Radom Varables of he Dscree Te r.v. X(s sace of X{:X(s, ss}.m.f.: P(Xf( such ha f(>, sum of all f(. Hergeomerc dsrbuo: ( ( ( f (,,,, µ, (selecg r objecs a radom whou relaceme from a se comosed of wo es of objecs Q: deerme c such ha f(*c was.m.f.,, : se f(+..+f(c+c+..+c55c, so c/55 rel. freq: (X/ hergeo-- defecve ems eacl : P(X, a mos : P(X+P(X. Mahemacal Eecao s E[ u( X ] u( f (, mea s E( X S µ oo, E[( X ] E( X Var( X f c s a cosa, E(cc c cosa ad u fuco, E[c u(x]ce[u(x] E[cu(X+cu(X] ce[u(x]+ce[u(x] Q: radom eger from frs : f(/, E[X(-X]E[X]-E[X^] use seres o g. o calculae E(X loo ckes: $.5 each,,6 draw ou of Ml,,-$5, 4-$,, -$5,, -$,: E[X](*5+..+*/Ml value for us: E[X]-$.5 E[X] s DE whe seres does o coverge E[( X ] skewess s, o ar s: E[( X ] E[ X ] µ E[ X ] + µ. Beroulls Trals ad he Bomal Dsrbuo Ber. Trals: wo dffere oucomes (es/o each eerme rob. of success, q- of falure Beroull Dsrbuo: f ( (,,, M ( + e, µ, (,,! Bomal Dsrbuo: f ( (,,..,, ( ( M + e, µ, (, X#!(! of success Beroull rals b(, Q: draw wo dff. color balls: 7Red, Whe, X f red, X f whe > Beroull ad 7/8 es w/ 6 Qs, 5 oss. awers: P(correc o # ad #4P(C,I,I,C,I,I(/5^*(4/5^4 P(correc o QsP*(6 choose whe ask wha s eeced value, lug ha umber o E[#] bomal: P( X 5 P( X 4 rob. of wg a leas oe rze b(# of ckes bough, rob. loo s P( X P( X use formula for f (.4 The Mome-Geerag Fuco m.g.f: ( ( X X M E e e f (, -h<<h, M '( e X f ( S S ( z z z z..., z M "( < <, µ,, M (E(X, M (E(X M e f X "( ( S

3 egave Bomal Dsrbuo: ( r r( µ r r r f ( (, r, r +,..., r ( e M (, l( r [ ( e ] >, (,, do Ber. rals ul r successes, X# of ral whe success observed Geomerc Dsrbuo: f ( ( e, M (, l( ( e <, µ,, X# of rals o oba s success seq. of Ber. rals m + m Uform Dsrbuo: f (,,,.., m, µ, m Q: whe asked o fd mea&ec kowg M(, check o see whch dsr. M( s for ease whe we have b(, wh >.5, se Y s b(,-, P(X<-P(Y< Geo: eole s brhdas (/65, P(X>4(-^4 P(X<P(X<99-(-^99 k + P(( X > k + j ( X > k P( X > k + j q j P( X > k + j X > k q P( X > j k P( X > k P( X > k q.5 The Posso Dsrbuo λ e λ rob. of oe chage erval legh h s h(/ f (, M ( e λ, µ λ, # of eves!` occurg a u erval, eves are occurg radoml a a mea rae of er u erval Q: P(<X<5P(X<5-P(X< P(X>-P(X<, P(X f( cus/hr, mea e. mea(flaws er u*legh 4. Radom Varables of Coous Te Prob. des. fuc. (.d.f.: f( sasfes: rob. of eve X ( e A s P( X A f ( d o S, elsewhere. [Cumlve[ dsrbuo fuco (c.d.f.: F( P( X f ( d, F (f(.d.f. P(Xb ad P(a<X<b a sg <+ F(b-F(a. µ E( X f ( d, Var( X E[( X ] ( f ( d, M ( e f ( d, h < < h π - (h ercele (area uder f( o he lef of π A π s : f ( d F( π (, Q: o fd Ds. Fuc, ake egral of f( f f ( {, oherwse, he F( {, (, < <, } fd c such ha f( was.d.f., se egral of f( from o c we usuall relace mus-f wh lower lm 4. The Uform ad Eoeal Dsrbuos Uform dsrbuo: F((-a/(b-a o a<b, o <a, o b,.d.f: f(/(b-a, ab sa, X s U(a,b b a a + b ( b a e e µ,, M (,, (selec a o a radom from [a,b] ( b a Eoeal dsrbuo: wag me for s arrval whe observg Posso rocess wh mea rae of chage(arrvals λ / θ : f ( e,, θ M (, <, µ θ, θ, M '(, θ θ θ θ ( θ θ / θ / θ M "(, F( { e o <,o <, meda m: m θ l(, P( X > e ( θ Q: U: P(X>8-P(X<8-F(8 (see s le P(<X<8F(8-F( cusomer arrves a cera wdow of me EXP: P(<X<F(-F(, P(X>-F(, P(X>4 X>P(X>-F( wag me ul s call arrves lamdacalls/mosso mea/legh

4 4. The Gamma ad Ch-Square Dsrbuos Gamma fuco: Γ ( e d, <, Γ ( (! Gamma dsrbuo: lke eoeal, bu wag ul h arrval. α / θ e f (, < <, α Γ( α θ α k λw M (, < ( λw e, ( θ α θ µ αθ, αθ, F( w k k! Ch-square dsrbuo w/ r degrees of freedom: gamma wh, / sum of squares of r deede (, r / / e varables. X s χ ( r. f (, <, M (, < r / / Γ( r / ( r, µ r, r, r / w/ w e dw, r, r / F( Γ( r / Q: Gamma ul h call arrves P(X<5F(5(use F(5 P(X>5- use ergral from 5 o + f f <5, use f o 5 or -P(X<5 (7 / e[ 7 /] (7 /[ e[ 7 /]] [e[ 7 /]] Ch-square use able f 5 observ., 4 degr. Freedom, fd rob. ha a mos of 5 obs. Eceed P(X< (able IV, so we have b(5,.9, add u for P(X+P(X+P(X+P(X 4.4 The ormal Dsrbuo Errors measuremes, heghs of chldre, breakg sreghs < µ <, <, M ( e( µ + /, µ E( X µ, Var( X, X s ( µ, w/ z e dw If Z s (,, Z has sadard orm. ds.: Φ ( z P( Z z, Φ( z Φ ( z π a X b b a P( a X b P( Φ( Φ ( b/c ( X / s (, If X s ( µ,, he V ( X / Z s χ ( e( ( u / f (, π 5. Dsrbuos of Two Radom Varables f(, P(X, Y jo.m.f. sasfes: f (, ad (, S P[( X, Y A] f (, (smlar for df, (, A ergrals from o + f Margal.m.f. of X: f( f (, P( X, S (same for Y X ad Y deede ff P(X, YP(XP(Y or f(,f(f(, oherwse deede (whe suor S s o recagular. Eeced value of u( X, X : E[ u( X, X ] u(, f (, E[ u ( X, X ] E( X µ, f ux varace of X s Hergeomerc dsrbuo for : Tromal dsrbuo: mea of X s (, S E[ u ( X, X ] E[( X ],, ( ( ( ( f (, (margal mf are regular hergeo. ds.! ( f (,!!(!, X s b(,, X s b(,, X & X deede

5 Q: Margals: of : (+/, of : (+/ P(X>Y(sum of all scearos where X>Y same for P(X+YP(X,Y+P(X, Y samle suor for hergeo (cards: S{(X,X <+<, <, <} romal: S{(, <, <7, +<7} ragular romal P(X<: use margal mf for X. 5. The Correlao Coeffce Covarace of X ad X: E[ u( X, X ] E[( X ( X ] Cov( X, X f Cov( X, X u( X, X ( X ( X Correlao coeffce of X ad X: ρ f sd. devs. > µ E( X f (, f (, f( Covarace: Cov( X, X E( X X µ ( ( f (, S Leas-squares regresso le: sloe ρ, he le s µ + ρ (, e. value of square s K b E Y b X, s m: K ( {[( µ ( µ ] } Samle Correlao Coeffce for Emercal Dsrbuo: r ( ρ ( ρ corr. coeff of wo de. varables s s, samle leas squares regresso le s ˆ + r ( s, E(XYsum of all combaos Q: Use boed formula for meas of ad ( f ( f ( of ad ad he fuco: **f(,+**f(,+**f(,+**f(, romal ds E(X ad E(Y: calculae f(, f(,, f(, f(,, he add o E(X ad E(Y resecvel, Var(X^*f(+..+^*f( mea-squared o ge corr. Coef. calc meas of ad, calculae E(XY, he Cov(X,Y, he lug he umbers 5. Codoal Dsrbuos codoal.m.f. of X, gve Y, s f (, g( f ( rovded f(> of Y, gve X, s f (, h( f ( cod. rob.: P( a < Y < b X h( cod. eecao: E[ u( Y X ] u( h( cod. mea { : a< < b} of Y, gve X, µ Y E( Y h( cod. var. of Y, Y E{[ Y E( Y ] } [ E( Y ] h(, also Y E( Y [ E( Y ] f E(Y s lear, he s E( Y µ + ρ ( (same as bes-f le For Tromal: X ad Y have margal bomal dsrbuos b(, ad b(, hus h( f (, (! f (!(!, hus codoal.m.f. of Y, gve X, s bomal b,, cod. meas E( Y ( ad E( X ( ρ ( ( Q: wegh of soa observed mes s b(,, so are ees of fl use E(X, Var(X(- formulas how dsrbued codoall fd h( or g( erms of or resecvell, f bomal, use formula o fd E(X^- XY+Y^, fd E(X, he E(Y, Var(X, Var(Y, E(X^E(X^+Var(X, calculae corr. coef. usg formula

6 seco, calculae E(XY, lug all #s. Le X have a dsr. ad le cod. dsr. of Y, gve X, be, he jo.d.f. s f(,h( *f(, samle E( X g( d (draw cs here 6. Ideede Radom Varables P(X ad XP(XP(Xf(f(jo.m.f. Eeced value of Yu(X, X: E[ u( X, X ] u(, f( f( g(, g(.m.f. of Y. S S S If X ad X de., e. value of roduc u(xu(x s roduc of e. Values of u(x ad u(x ad E[( X ( X ] E( X E( X If r.v. s.. have same.m.f., he we sa radom samle sze? from a dsrbuo w/.m.f. f( jo.m.f. s f(f(. Q: P(Xa,Xbf(af(b f P(X+Xa, sum he ossble combos,.e. P(X, X5+P(X,X4 f raged ad df, do ergrals each over rages f E( X X... f ( d f ( d E( X E( X f P(ma X < a P( X < a... P( X > a oa ob E( X µ + 6. Dsrbuos of Sums of Ideede Radom Varables Covoluo formula: g( P( Y f ( k f ( k k Thm: X, X, X de. r. v.s w/ jo mf f(*..*f( le Yu(X, X,..,X have mf g(, he E( Y g(... u(,,.., f ( f (... f ( (for co. e, egrals relace Sum Thm: If X, X, X de. r. v.s he E[ u ( X u( X... u( X ] E[ u ( X]... E[ u( X ] Thm: Mea ad Var. of Y a X where a,..a are real cosas, are µ a µ ad a X + X X Samle mea: X (he a/ Thm: If X, X X de. r. v.s wh M (, he m.g.f. of Y a X s Y Y X Y M ( M ( a Cor: If X, X.., X observaos of a radom samle wh m.g.f. M(, he (a he m.g.f. of Y X s ( [ ( ] M Y M (b m.g.f. of (/ X X s M ( [ M ( ] X Q: fd mea E(X he E(X^, lug Y-X+X, µ, µ 7, 9, 5, he E(Y-*+7 Var(Y(- ^*9+(^*5 o show whch dsr., comue M( usg heorem.

7 6. Radom Fucos Assocaed wh ormal Dsrbuos Thm: If X, X,..X are observaos of a radom samle of sze from he ormal dsrbuo ( µ,, he he dsrbuo of samle mea (/ X X s ( µ, / Thm: Le de. dsrbuos X, X X be χ ( r, he YX+ +X s χ ( r + r r Thm: Le de. Z, Z Z have sad. orm. ds. (,. The WZ^+..+Z^ s χ ( Thm: If X, X, X are observaos from a radom samle of sze from ormal dsr. ( µ,, (/ X X ad S (/ ( X X, he (a X ad S are deede (b W ds. ( S ( X X ( X X s ( µ, c c s χ (. OTE: samlg from orm. dsr. U ( X s χ ( Thm: If X, X X are muuall de. orm. var., he. oe: f YX-X, he Y s ( µ, + Y χ ( ad c X has orm. a X b b a Q: fgure ou mea ad var ad use P( a X b P( Φ( Φ ( Soa boes: X wegh of bo, X s (6.5,.4, 9 boes radom, how ma < 6.7? A: W. W s b (9,.5 ad ( ( ( P( W (.95 + (.5(.95 + (.5 (.95 Zs, Ws, radom samle do ch-square hm whe chsquare, fd whch of wo forms s, look u values able le S^ be samle varace of 9 weghs meas s χ (8 fd P(X>X: se YX-X, so P(Y>, use oe o ge values for orm. ds., he do as usual. 6.4 The Ceral Lm Theorem Mea X of a radom samle of sze from a dsr. wh mea µ ad var. > s a rad. var. wh: ( E X µ ad Var( X CLThm: If X s he mea of a radom samle X, X X, of sze from a dsrbuo wh X a fe mea µ ad a fe varace >, he he dsrbuo of W / as. So, P( W w Φ ( w ad X s ar. (, / µ ad Y s ar. ( µ, Q: fd mea E(X ad E(X^ use mea or sum formula as arorae YX+ +X s X s (, he lm χ (