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Binomial Distribution: Must: fxed trials, two outcomes(success or ailure), P(success)=p and P(ail)=1-p, and trials areindependent/ with replacement ater each trial. !"#"!$%&'P*!*%"%$+

P (x )= n !

x ! (nx ) !p

x (1p )nx ; X= 0,1,2,3

=np;2=np (1p ); =np(1P) ounts the numer o successes in the n trials, and is

called the binomial random variable= discrete ariale

x. Pat has ui0 with 1 M uestions. ach uestion hasfe possile answers. 2hat is the proailit3 that Pat 4etsno answers correct5

&=1, P=1/6 (1-p)=7/68 Pat 4ettin4 no answers ri4ht

P(x=), P (0)= 10 !

0 ! (100 )!(0.2)0 (10.2)100

Cumulative problems where it includes multiple xalues P(9n), so what is the proailit3 Pat will ail5 $hatmeans 7 uestions or less were answered so

P (X 4 )=p (0 )+P (1 ) +P(4) Using Binomial Tablen=1,

P (X k) ,thusk=4,P=0.2 ;ast wa3 to calculateP(9=n) usin4 tale:

P (X=2 )=, p (X 2 )P(X 1)

P (X K)=1P(X (k1 ))

oisson distribution-$he numer o successul eents thatoccur in a time interal is independent o the numer osuccess that occurs in another time interal. %t fts cases orare eents that occur oer a fxed amount o time or within aspecifed re4ion. !

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6)=14= 146 14=.?-=.??) =34= 346 34=.H-.H=

7eometri# 8andom 9ariable 6$he numer o independent*ernoulli trials until te :rst su##ess i 3ou repeat anexperiment and it has D outcomes, calculates numer o trailsuntil 3ou 4et frst success. $he proailit3 o success, p,isconstant. $he proailit3 mass unction, pm, is:

1px1p , E (x )==1

p, V(x )=2=

1p

p2

f(x)= x. omeone is tr3in4 to taAe the road test. % the

proailit3 o passin4 the test is 7J, what is the

proailit3 that this person willpassthe test at second

shot510.4 210.4

f(2)= imilarities+ di 7eometri#

random variable imilarities: independent trials,proailities o success / ailure Diow man3 possile wa3sH students can e seated around a circular tale5&61( = Ea) i two students ant to sit side 3 side5 &E61(2 = F@2) % two do not ant toside-3-side &61( 6&E61(2 = E6F@2

8andom 9ariable$ ! random ariale is a unction or rulethat assi4ns a numer to each outcome o an experiment

Dis#rete 8andom 9ariable one that taAes on a #ountablenumer o alues (D,@). %nte4ers are discrete

Continuous 8andom 9ariable one whose alues are notdiscrete, not countale .4. $he minimum alue lon4er than@minutes (@.,or@.,or@.) eal BKsrobabilit" distributions a tale, ormula, or 4raph thatdescries the alues o a random ariale and the proailit3associated with these alues can e: 1( Dis#rete8andom 9ariable6

0 P(X) 1 Gor all ' and P (x )=1 so it isrelatie reuenc3, x. at least one teleision ut no more thanthree P(1 Q 9Q @) = P(1) G P(D) G P(@) and relatie proailit3is 3 diidin4 B o households/ total households2( Continuous 8andom 9ariable6

robabilit" Distribution6 $here is DJ chance o closin4 asale on each call.2hat is the proailit3 distriution o thenumer o sales i callin4 three customers5 E!2 $

P (%|& )=P (%& )('oint pro(a(i$it! )P ( & )(mar)ina$ pro(a(i$it! )

P (%|& )=P (%& )

P ( & )1. omplement o ent &?C ( = 1 * &?(D. #nion o ents &? or B( = &?( - &B( * &? and B(3@ %ntersection o ents &? and B(= &? H B( I &B(@ Mutuall3 xclusie ents &? or B( = &?( - &B(6. % we multipl3 oth sides o the euation 3 P(*) we hae:

P(! and *) = P(! R *) S P(*)F@ ! special case: ? and B are independent, then &? and

B( = &?( I &B(

Jas oG 'pe#ted1.(=#the expected alue o a constant (c) is Nust the alueo the constant itsel.x. (x)=(@xG1)= @O(x) G1D. &X-#(=&X(-#@. (=#&X( 2e can UpullV a constant out o the expectedalue expression(either as part o a sum with a randomariale 9 or as a coeWcient o random ariale 9)'ample@ Monthl3 sales hae a mean o XD6, and astandard deiation o X7,. Multipl3in4 sales 3 @J andsutractin4 fxed costs o X?, calculate profts.1)Eescrie the prolem statement in Math:-sales hae a mean o XD6,(ales) = D6,-Profts is (Proft) = (.@(ales) C?,)= Y.@(ales)Z C?,= .@(ales) C?,= .@(D6,) C?,= 1,6

Jas oG 9arian#e

1@9(=0$he ariance o a constant (c) is 0ero.2@9&X-#(=9&X(

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$he ariance o a random ariale and a constant is Nust theariance o the random ariale@.9(=#K29&X(

$he ariance o a random ariale and a constant coeWcient isthe coeWcient suared times the ariance o the randomariale.