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DISCRETE & RANDOMDISCRETE & RANDOM

VARIABLESVARIABLES

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Discrete random variableDiscrete random variable

 A discrete random variable is one wic ma! ta"e on A discrete random variable is one wic ma! ta"e ononl! a co#ntable n#mber o$ distinct val#es s#c as % 'onl! a co#ntable n#mber o$ distinct val#es s#c as % '( ) * +++( ) * +++

Discrete random variables are #s#all! ,b#t notDiscrete random variables are #s#all! ,b#t notnecessaril!- co#nts+ I$ a random variable can ta"e onl! anecessaril!- co#nts+ I$ a random variable can ta"e onl! a$inite n#mber o$ distinct val#es ten it m#st be discrete+$inite n#mber o$ distinct val#es ten it m#st be discrete+

E.am/les o$ discrete random variables incl#de teE.am/les o$ discrete random variables incl#de te

n#mber o$ cildren in a $amil! te 0rida! ni1tn#mber o$ cildren in a $amil! te 0rida! ni1tattendance at a cinema te n#mber o$ /atients in aattendance at a cinema te n#mber o$ /atients in adoctor2s s#r1er! te n#mber o$ de$ective li1t b#lbs in adoctor2s s#r1er! te n#mber o$ de$ective li1t b#lbs in abo. o$ ten+bo. o$ ten+

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Relative $re3#enc!Relative $re3#enc!

Te onl! wa! to 1et an estimate o$ te /robabilit! is toTe onl! wa! to 1et an estimate o$ te /robabilit! is to

trow te stic"s man! times and $ind wat we call te4trow te stic"s man! times and $ind wat we call te4  5 relative $re3#enc!relative $re3#enc! 

 5 6 Can be di$$erent eac time w!76 Can be di$$erent eac time w!7

 5 I$ te stic" is trown '%% times and te res#ltsI$ te stic" is trown '%% times and te res#ltsare4are4

 5 0lat side #/4 8% 1ivin1 an estimate /robabilit! o$ %+80lat side #/4 8% 1ivin1 an estimate /robabilit! o$ %+8

 5 C#rved side #/4 )% 1ivin1 an estimate /robabilit! o$ %+)C#rved side #/4 )% 1ivin1 an estimate /robabilit! o$ %+)

N#mber o$ $lat sides #/+N#mber o$ $lat sides #/+ %% '' (( )) **ScoreScore 99 '' (( )) **

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Listin1 /ossibilities4Listin1 /ossibilities4

ave a s!stemave a s!stem

$ $ 

cc

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cc $ $ 

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$ $  cc cccc $ $  $ $ 

cc $ $  cc

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cc cc cc

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$ $  $ $  $ $  cc

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cc $ $  $ $  $ $ 

cc $ $  $ $  cc

cc $ $  cc $ $ 

cc $ $  cc cc

cc cc $ $  $ $ 

cc cc $ $  cc

cc cc cc $ $ 

cc cc cc cc

' stic"

( stic"s

) stic"s

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:or"in1 o#t /robabilities4:or"in1 o#t /robabilities4

$ $ 

cc

$ $  $ $ 

$ $  cc

cc $ $ 

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$ $  $ $  cc

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$ $  $ $  $ $  cc

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$ $  cc $ $  cc

$ $  cc cc $ $ 

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cc $ $  $ $  $ $ 

cc $ $  $ $  cc

cc $ $  cc $ $ 

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cc cc cc cc

' stic"

( stic"s

) stic"s

Each stick falls independently,

f=0.7 c=0.3

1 x P(f,f,f,f) = 0.7 x 0.7 x 0.7 x 0.7= 0.2401 = 0.74

4 x P(f,f,f,c)= 0.73 x 0.3 x 4=

6 x P(f,f,c,c)=0.72 x 0.32 x 6 =

4 x P(f,c,c,c) = 0.7 x 0.33 x 4 =

1 x P(c,c,c,c) = 0.34 =

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Relative $re3#enc!Relative $re3#enc!

Te score is an e.am/le o$ discrete random variable+Te score is an e.am/le o$ discrete random variable+

 5 Let S stand $or score+ Ca/ital letters are #sed $or randomLet S stand $or score+ Ca/ital letters are #sed $or randomvariables+variables+

 5 ;,S %+*''>%+*''> %+(*%'%+(*%' %+%%?'%+%%?'

Note s is #sed $or individ#al val#es o$ te random variable S

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;,@

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TASTAS

E.ercise A 5 ;a1e 9) & 9*E.ercise A 5 ;a1e 9) & 9*6#estions4 ' ( ) 9 & >6#estions4 ' ( ) 9 & >

Do rest at ome+Do rest at ome+

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Mean variance and standardMean variance and standard

deviationdeviation

I$ I were to trow '%%%% times I co#ld wor" o#t teI$ I were to trow '%%%% times I co#ld wor" o#t te

mean li"e te below+ M#lti/l! eac o$ m!mean li"e te below+ M#lti/l! eac o$ m!

/robabilities b! '%%%% and ten divide b! '%%%%/robabilities b! '%%%% and ten divide b! '%%%%

s+s+ '' (( )) ** 99

;,S %+(>*>%+(>*> %+*''>%+*''> %+(*%'%+(*%' %+%%?'%+%%?'

84.210000

8152401441163264627561=

×+×+×+×+×=mean

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Mean variance and standardMean variance and standard

deviationdeviation

owever m#lti/l!in1 and dividin1 b! '%%%%owever m#lti/l!in1 and dividin1 b! '%%%%

bot to/ and bottom seems #nnecessar!bot to/ and bottom seems #nnecessar!

and it isand it is

s+s+ '' (( )) ** 99

;,S %+(>*>%+(>*> %+*''>%+*''> %+(*%'%+(*%' %+%%?'%+%%?'

84.20081.052401.044116.032646.020756.01   =×+×+×+×+×=mean

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MEAN o$4MEAN o$4

Discrete random VariablesDiscrete random Variables

Mean S

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VARIANCEVARIANCE

Mean < %Mean < %..%+'9 G '.%+(9 G (.%+(9 G%+'9 G '.%+(9 G (.%+(9 G

).%+(9 G *.%+'' %+'9%+'9 %+9*'9%+9*'9

'' %+H%+H %+?'%+?' %+(9%+(9 %+(%(9%+(%(9(( %+'%+' %+%'%+%' %+(9%+(9 %+%%(9%+%%(9

)) '+''+' '+(''+(' %+(9%+(9 %+)%(9%+)%(9

**

(+'(+'

*+*'*+*'

%+'%+'

%+**'%%+**'%

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Variance & Standard deviationVariance & Standard deviation

.. .. ,,.-.-(( ;,@

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TASTAS

;a1e 9> 3#estion (;a1e 9> 3#estion (

omewor" 5 test !o#rsel$ omewor" 5 test !o#rsel$