04 discrete probability distribution mmc_.pdf
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Math801MathematicalMethodsforComputing
DiscreteProbabilit Distribution
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ConceptofRandomVariable
DiscreteProbabilityDistribution
TheBinomial
Distribution
Home
Work
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Conceptof
RandomVariable
Inmostprobabilityproblems,weare
interestedin
one
number
that
is
associated
withtheoutcomeofexperiment.Such
,
outcomeof
experiment,
is
called
Random
.
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Examplesof
RandomVariable
Problems
1. Inmanufacturing,x=No.ofdefectiveitems
= .
,
3. Throwingtwodice,z =sumoftwodice
situationsinvolvingrandomness,therefore,
a o ea ovevar a ex,y,an zare reedifferentrandomvariables.
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StudyingRandomVariablesweinterestin:
valuesintheirrange.T is
sprea
o
pro a i ities
or
t e
various
valuesofrandomvariablesiscalleda
probabilitydistribution.Inthefollowing,weconstructanexample:
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Experimentof
Throwingtwo
Dice
Results:
Tabulated:
Thinkabouthavinga
6
sumo wo ce=z=
P(X=7)=6/36
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Pro a i ity istri ution unction p ort esum
oftwo
dice
7
Note:Abovediagramisexperimental,makeyourselfidealpdf.
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distribution
calledprobability
densityfunction
orprobability
massfunction.
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Whentherandomvariabletakesthediscrete
values,its
robabilit
distribution
will
be
DiscreteProbabilityDistribution.
,
defectiveitemsineachshipment willhave
discreteprobabilitydistributions.
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Byunderstandingtheunderlyingpattern
behindeach
ex eriment,
we
have
formulated
differentprobabilitydistribution.
,
importantdiscreteprobabilitydistribution
calledTheBinomialProbabilityDistribution.
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Considerexperimentoftossingacoin:
.
2. Successprobability,p,remainssame
3. Thesuccessive
trials
are
all
independent
threeconditionsarecalledBernoullitrails.Theexperimenthavingn Bernoullitrialsis
calledabinomialprobabilityexperiment.11
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WhenXdenotesthenumberofsuccessesinn
trialsof
abinomial
robabilit
ex eriment,
it
iscalledabinomialrandomvariablehaving
. . .
X, thebinomial
p.d.
is
given
by
the
formula:
( ; , ) ( ) , 0,1,2,...,x n x
n
b x n p P X x p q x n
-
= = = =
where q =1-p is
the
probability
of
failure 12
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Q:Acoinistossed5times.Whatareprobabilities
ofobtainin
various
number
of
heads?
SincetheseareBernoullitrialswithn=5times.
. .
withp=1/2,q=1/2,andn=5.Thereforeweapply
x n xn - , , , , ,...,x
= = = =
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x n xn
-
MainformulaforBPD
, , , , ,...,x
551 1 1
( ; 5 , ) ( ) , 0,1,2, 3, 4, 5
x x
b x P X x x x
- = = = =
0 5 051 1 1 1 1- ; ,
02 2 2 32 32= = = = =
51 1 1 1 1 10(3; 5 , ) ( 3) 10
32 2 2 8 4 32b P X
- = = = = =
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.
MakeaMatlab programtoimplement
( ; , ) ( ) , 0,1,2,...,x n xn
b x n p P X x p q x n x
- = = = =
suchthat,
for
any
input
of
nand
p,
program
enerateahisto ram.
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Questions?
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