Discrete Random Variables.ppt
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Transcript of Discrete Random Variables.ppt
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8/19/2019 Discrete Random Variables.ppt
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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
$ $ $ $
$ $ cc
cc $ $
cc cc
$ $ $ $ $ $
$ $ $ $ cc
$ $ cc $ $
$ $ cc cccc $ $ $ $
cc $ $ cc
cc cc $ $
cc cc cc
$ $ $ $ $ $ $ $
$ $ $ $ $ $ cc
$ $ $ $ cc $ $
$ $ $ $ cc cc
$ $ cc $ $ $ $
$ $ cc $ $ cc
$ $ cc cc $ $
$ $ cc cc cc
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 $ $
cc cc
$ $ $ $ $ $
$ $ $ $ cc
$ $ cc $ $
$ $ cc cc
cc $ $ $ $
cc $ $ cc
cc cc $ $
cc cc cc
$ $ $ $ $ $ $ $
$ $ $ $ $ $ cc
$ $ $ $ cc $ $
$ $ $ $ cc cc
$ $ cc $ $ $ $
$ $ cc $ $ cc
$ $ cc cc $ $
$ $ cc cc cc
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
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$
