Lecture 2-Taburan Binomial
Transcript of Lecture 2-Taburan Binomial
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SPECIAL PROBABILITYDISTRIBUTION IBINOMIAL DISTRIBUTION
JUHAIDAH BT JAMALFACULTY OF TECHNICAL
AND VOCATIONALEDUCATIONUTHM
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TABURANKEBERANGKALIAN?
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Experiment:
Toss a coin 3 times.Observe the number of
heads. The possible results
are: zero heads, one head,
two heads, and three heads.
What is the probability
distribution for the number
of heads?
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Probability Distribution of Number ofHeads Observed in 3 Tosses of a Coin
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Random Varia!"#
Random $aria!" - a %&an'i'(r"#&!'in) *rom an "+,"rim"n''-a'. ( /-an/". /an a##&m"
di0"r"n' $a!&"#1
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T(," o* Proai!i'(Di#'ri&'ion
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F"a'&r"# o* a Di#/r"'"
Di#'ri&'ion
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Te main features of a dis!rete"robability distribution are#
T-" #&m of te "robabilities of tevarious out!omes is 2133$Te "robability of a "arti!ular
out!ome is "'4""n 3 and 2133$
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BINOMIAL DISTRIBUTION%inomial distribution is a basi! distribution
tat models !an!e variation in repetitionsof an experiment that has only two possibleoutcomes$
&
E+,"rim"n'
Fai!&r"S&//"##
p q
Probability :p + q = 1
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'(am"le %inomial'("eriment#
)en a !oin is tossed* it !an land -"ad# or'ai!#$
)en a baby born* it +ill be eiter ma!" or
*"ma!"$,n a basetball .ame* a team eiter 4in or
!o#"#$
/ medi!al treatment !an be !lassi0ed as
"0"/'i$" or in"0"/'i$"* de"endin. onte results$
/ "erson !an be !lassi0ed as avin.norma! or anorma! !ood ,r"##&r"*de"endin. on te results$
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TABURAN BINOMIAL
S&a'& &5i 6a5i Binomia! ada!a- #&a'& &5i6a5i(an) m"m,&n(ai /iri7/iri "ri6&' 8
iai itu terdiri dari"ada sebilan.an n ,"r/&aan tabersandar$
emua !ubaan adala merdeatida bersandar #a'&/&aan 'ida6 #a!in) m"m,"n)ar&-i /&aan !ain$
etia" !ubaan men.asilan sala satu dari"ada 2esudaan yan. disebut 96"5a(aan9dan 96")a)a!an9$
K"aran)6a!ian 6"5a(aan "r!a6& da!am #a'&
,"r/&aan ia!a- , dan nilai " adala teta" dari satu"er!ubaan e"ada satu "er!ubaan7
K"aran)6a!ian 6")a)a!an "r!a6& ia!a- q = 1- p
Pemboleuba ra+aXia!a- i!an)an 6"5a(aan (an)"r!a6& da!am n,"r/&aan1
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No'a'ion *or '-" Binomia!Proai!i'( Di#'ri&'ion
P (S) # Te symbol for te "robability of
su!!ess
P (F) # Te symbol for te "robability of
failurep # Te numeri!al "robability of a
su!!ess
q # Te numeri!al "robability of a
failure# P(S) = p and P(F) = 1 - p = q
n # Te number of trials
X # Te number of su!!esses in n trials
Note that 0 X n and X = 0, 1, 2, 3, .. n.18
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E+am,!" :12
Consider te e("eriment !onsistin. ;*'""n'o##"d o* a /oin$
D"'"rmin" i* i' i# a inomia! "+,"rim"n'1
So!&'ion8
1$ Tere are a total of 15 trials 9tosses:# ni# '-" n&m"r o*11
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Ta6ri* Ta&ran Binomia!
Taburan Kebaran!alian "inomial ditulis sebaai
X ~ B (n,p)
diba#a sebaai :
"Xmempunyai taburan Binomial dengan
parameterparameter ndanp"
X = number of su!!esses in ntrials
n$ number of trials
p$ probability of su##ess on sinle trial
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i i ! i!i
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Binomia! Proai!i'(Form&!a
P%& $ r' =1: = C 9-' %-' $ -
P9>=2: = C 9-' %-' $ +
!
3 !
!
#
!
!
2 2
3
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more easier $'
imple $'
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E>AMPLE :1 PG1 =
/$ no -"adP(X=0)
B1 "+a/'!( on" ead P(X=1)
C1 *"4"rtan t+o eads P(X'2)
D1 mor" '-an tree eads P(X3)
E1a' !"a#' tree eads P(X3)
/ #oin is tossed fi$e times. 0et & be a random variable for number ofhead is re#orded. ind the probability of ettin
%et2 probability of su##ess %head',
p $ -2 Probability of failure %tail',3 $ -
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E>AMPLE :1 PG1 =
A1 a' mo#' one ead P(X1)
B1 no' mor" '-an t+o eads P(X 2)
C1 on" 'o '-r"" eads P(1X3)D1 B"'4""n '4o and ;$" ead P(2'X')
E1 T4o &' !"## tan four eads P(2X'*)
/ #oin is tossed fi$e times. 0et & be a random variable for number ofhead is re#orded. ind the probability of ettin
%et2 probability of su##ess %head',p $ -
2 Probability of failure %tail',
3 $ -
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E+"r/i#" 28
Tere are ;$" Bi.ts daily from Pittsbur.via /ir+ays into te %radford*Pennsylvania* e.ional /ir"ort$ u""ose te"robability tat any Bi.t arrives late is313$ )at is te "robability tat non"ofte Bi.ts are late today
9/ns+er#8$32&&:
E+"r/i#" 8/ study in Eune 2883 by te ,llinoisDe"artment of Trans"ortation !on!ludedtat 1 "er!ent of front seat o!!u"ants
used seat belts$ / sam"le of 2vei!les is1
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E+"r/i#" :
P"ra' 6"!&!an murid dalam satu"e"erisaan iala =$ Tentuanebaran.alian baa+a d&a dari,ada#"mi!an m&ridyan. men.ambil
"e"erisaan tersebut a6an )a)a!da!am ,","ri6#aan '"r#"&'1
Ja4a,an8 31=
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Ho4 To U#"Binomia!Proai!i'( Ta!"
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/ltou. te binomial
distribution is easily evaluatedby !al!ulators*
ometimes it is more!onvenient to use binomial"robability table es"e!ially
+en te times of trials +o%ethan ten (10).
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T&' T* +
i. X ~ B (!, .-), ind /(X01)
i. X ~ B (#, .2), ind /(X0#) .445
.51-#
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T,P TO '
%,NOF,/;PO%/%,;,TG
T/%;'
Here aresome ti"s
o+ to!om"ute te"robability by
usin. table22
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E+am,!" :1 PG
a P+ 2= = 8$468
P+ 23 = 2 P + 22
= 1 8$& = 8$8822
/ P+ 3 = P> 2 = 8$438
d P> 2 P> 2:
= 8$43 - 8$6& = 8$88&6
" P3 > =P> 3 7 P>
= 8$5535 - 8$88&6= 8$545
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4 $ 56
P $ 6.7
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E>AMPLE :1 P)1
/ ne+ medi!ine as been tasted and :=from te "atients +ose tae tatmedi!ine be!ame ealty$ ,f '"n"atients
are sele!ted at random from .eneralos"ital in order to tae tat medi!ine*0nd te "robability tat
a '4o ,a'i"n'# "/om"-"a!'-(
a' !"a#' #i+ ,a'i"n'# 4i!! "-"a!'-(
/ no' mor" '-an ;$" ,a'i"n'#
ar" -"a!'-(2
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*olution+
2
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E+"r/i#" 8
:=dari"ada !alon-!alon yan. men.ambilsuatu "e"erisaan '"!a- )a)a!$ Den.anmen..unaan adual* !ari ebaran.alianbaa+a dalam satu elas 233 oran)m&rid yan. men.ambil "e"erisaan ini*
9a: seuran.-uran.nya seten.adari"ada elas itu tela .a.al$An#4"r8 31332=
9b: tida lebi dari"ada 35 oran. muridtela .a.al$ An#4"r8 31==
9!: di antara 28 dan 35 oran. murid tela.a.al$
An#4"r8 312
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E+"r/i#" =8
eoran. "ena.a ereta menda"ati baa+asatu dari"ada setia" lima bua ereta yan.diletaan di tem"at ereta a.aannyaadala ereta luar bandar$ Pada satu ari18 bua ereta diletaan di situ$ Cariebaran.alian baa+a#
9a: seuran.-uran.nya 2 bua eretaluar bandar diletaan di tem"atleta ereta "ada ari itu$
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9b: lebi dari"ada 2 teta"i uran.dari"ada 5 bua ereta luar bandar
diletaan di tem"at leta ereta "adaari itu$ 31
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M"an and $arian/" o*Binomia! Di#'ri&'ion
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E>ERCISE 23
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E>ERCISE 723B!&man
6: / !oin is tossed 4 times$
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: Te Statistical Bulletin"ublised by
Fetro"olitan ;ife ,nsuran!e Co$ re"ortedtat 2I of all /meri!an birts result int+ins$ ,f a random sam"le of 888 birtsis taen* 0nd te mean* varian!e* and
standard deviation of te number of birttat +ould result in t+ins$232=121=
18:/ die is rolled 48 times$
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6ont7..