97'-7)1;=&$17'1'/0'/01)2'18'5)1a#a-:-2'#7;'%#$%$&%f'g...
TRANSCRIPT
© Joseph J. Nahas 2012 10 Dec 2012
!"#$%$&%'(#)"'**'+',#%-&'./01)2''
!"#$%&'!(')*&*#'
+,-.$/#-01'"2')"0/$'3*4$'
3'© Joseph J. Nahas 2012
10 Dec 2012
405#)"607"'18'955:-0;'#7;'<165="#$17#:'>#"/06#$&%'#7;'!"#$%$&%'?9<>!@'
• 9<>!'&1=)%0%'"/#"'6#2'A0'=%08=:'– 9<>!'BCDDCE'()1A#A-:-"2'#7;'!"#$%$&%'
! 97'-7")1;=&$17'"1'"/0'"/01)2'18'5)1A#A-:-"2'#7;'%"#$%$&%F'G-"/'#55:-&#$17%'"1'"/0'&165="0)'%&-07&0%'#7;'07H-700)-7H'
– 9<>!'BCICCE'!"#$%$&#:'>0"/1;%'J'4#"#'97#:2%-%'*'! *7")1;=&$17'"1'%"#$%$&#:'60"/1;%'G-"/'#7'065/#%-%'17'#7#:2%-%'18';#"#'
B'© Joseph J. Nahas 2012
10 Dec 2012
(15=:#$17'K0)%=%'!#65:0'• <17%-;0)'L4'M)0%/6#7'!9.'!&1)0%N'
– O0::';0P70;'515=:#$17'
– O0'&1=:;'1A"#-7'#::'"/0'3C3B'M)0%/6#7')0&1);%'#7;';0"0)6-70'"/0'%"#$%$&%'81)'"/0'8=::'515=:#$17E'! Q'R'"/0'>0#7'!9.'%&1)0'
! S'T''"/0'!"#7;#);'40K-#$17'18'"/0'!9.'%&1)0%'– O0'&1=:;'1A"#-7'#'%#65:0'18'%#2'UCC'M)0%/6#7')0&1);%'#7;';0"0)6-70'
0%$6#"0%'81)'"/0'%"#$%$&%E''! 6'1)'V'R'W%$6#"0;'>0#7'1)'9K0)#H0'!9.'%&1)0'-7'"/0'%#65:0'
! %'R'"/0'W%$6#"0'18'"/0'!"#7;#);'40K-#$17'
D'© Joseph J. Nahas 2012
10 Dec 2012
L1"#$17'
>0#%=)0' (15=:#$17'X)00Y'Z0[0)%'
!#65:0'\16#7'Z0[0)%'
Z1&#$17' >0#7' Q' W%$6#"0'18'"/0'>0#7F'9K0)#H0'
6'V'
!5)0#;' ]#)-#7&0' S3' !#65:0']#)-#7&0'
%3'
!"#7;#);'40K-#$17'
S' !#65:0'!"#7;#);'40K-#$17'
%'
<1))0:#$17' <1))0:#$17'<10^&-07"'
_' !#65:0'<1))0:#$17'<10^&-07"'
)'
`'© Joseph J. Nahas 2012
10 Dec 2012
!"#$%$&'a=":-70'UE ,#&YH)1=7;N'
9E O/2'!"=;2'!"#$%$&%'#7;'!"#$%$&#:'WV50)-607"#:'40%-H7b',E \080)07&0%'
3E ,#%-&'!"#$%$&#:'./01)2'9E ,#%-&'!"#$%$&#:'40P7-$17%'
-E 4-%")-A=$17%'--E !"#$%$&#:'>0#%=)0%'---E *7;0507;07&0c40507;07&0'
#E <1))0:#$17'<10^&-07"'AE <1))0:#$17'<10^&-07"'#7;']#)-#7&0'&E <1))0:#$17'WV#65:0'
,E ,#%-&'4-%")-A=$17%'-E 4-%&)0"0'K%E'<17$7=1=%'4-%")-A=$17%'--E ,-716-#:'4-%")-A=$17'---E L1)6#:'4-%")-A=$17'-KE ./0'<07")#:'Z-6-"'./01)06'
#E 40P7-$17'AE 4-&0'#%'#7'0V#65:0'
I'© Joseph J. Nahas 2012
10 Dec 2012
!"#$%$&'a=":-70'?&17"E@'BE X)#5/-&#:'4-%5:#2'18'4#"#'
9E d-%"1H)#6',E ,1V'(:1"'<E L1)6#:'()1A#A-:-"2'(:1"'4E !&#[0)'(:1"'WE >#"Z#A'(:1e7H'
DE <17P;07&0'Z-6-"%'#7;'d251"/0%-%'.0%$7H'9E !"=;07"f%'"'4-%")-A=$17'
-E O/1'-%'g!"=;07"h'--E 40P7-$17%'
,E <17P;07&0'Z-6-"%'81)'"/0'>0#7'<E Wi=-K#:07&0'18'"G1'>0#7%'
I'
j'© Joseph J. Nahas 2012
10 Dec 2012
!"#$%$&'a=":-70'UE ,#&YH)1=7;N'
9E O/2'!"=;2'!"#$%$&%'#7;'!"#$%$&#:'WV50)-607"#:'40%-H7b',E \080)07&0%'
3E ,#%-&'!"#$%$&#:'./01)2'9E ,#%-&'!"#$%$&#:'40P7-$17%'
-E 4-%")-A=$17%'--E !"#$%$&#:'>0#%=)0%'---E *7;0507;07&0c40507;07&0'
#E <1))0:#$17'<10^&-07"'AE <1))0:#$17'WV#65:0'&E <1))0:#$17'<10^&-07"'#7;']#)-#7&0'
,E ,#%-&'4-%")-A=$17%'-E 4-%&)0"0'K%E'<17$7=1=%'4-%")-A=$17%'--E ,-716-#:'4-%")-A=$17'---E L1)6#:'4-%")-A=$17'-KE ./0'<07")#:'Z-6-"'./01)06'
#E 40P7-$17'AE 4-&0'#%'#7'0V#65:0'
k'© Joseph J. Nahas 2012
10 Dec 2012
,#%-&'!"#$%$&#:'40P7-$17%'• 4-%")-A=$17N'
– ./0'5#[0)7'18'K#)-#$17'18'#'K#)-#A:0E''– *"')0&1);%'1)';0P70%'"/0'7=60)-&#:'K#:=0%'18'"/0'K#)-#A:0'#7;'/1G'
1l07'"/0'K#:=0'1&&=)%E''– 9';-%")-A=$17''A0';0%&)-A0;'A2'%/#50F'&07"0)F'#7;'%5)0#;E'
• ]#)-#A:0'R'VN'– 9'&/#)#&"0)-%$&'"/#"''#%%=60';-m0)07"'K#:=0%E'
• \#7;16']#)-#A:0N'– 9'8=7&$17'"/#"'#%%-H7%'#'7=60)-&#:'K#:=0'"1'0#&/'51%%-A:0'1="&160'18'
%160'150)#$17c0K07"E'
• (15=:#$17N'– ./0'"1"#:'#HH)0H#"0'18'1A%0)K#$17%'"/#"'&17&05"=#::2'6-H/"'1&&=)'#%'#'
)0%=:"'18'50)81)6-7H'#'5#)$&=:#)'150)#$17'-7'#'5#)$&=:#)'G#2E'– ./0'=7-K0)%0%'18'K#:=0%E'– M-7-"0'1)'*7P7-"0' P. Nahas
n'© Joseph J. Nahas 2012
10 Dec 2012
,#%-&'40P7-$17%'?&17"E@'• !#65:0N'
– 9'&1::0&$17'18'%160'18'"/0'1="&160%F'1A%0)K#$17%F'K#:=0%'8)16'"/0'515=:#$17E'
– 9'%=A%0"'18'"/0'515=:#$17E'• \#7;16'!#65:0N'
– W#&/'606A0)'18'"/0'515=:#$17'/#%'#'0i=#:'&/#7&0'18'A0-7H'&/1%07'#%'#'606A0)'18'"/0'%#65:0E'
• ,-#%N'– ./0'"07;07&2'"1'8#K1)'"/0'%0:0&$17%'18'=7-"%'G-"/'&0)"#-7'
&/#)#&"0)-%$&%E'• 9&$K0'4#"#'<1::0&$17N'
– (:#770;';#"#'&1::0&$17'G-"/'%50&-P&'H1#:%'-7'6-7;'"1'6#V-6-o0'"/0'-781)6#$17E'
• (#%%-K0'4#"#'<1::0&$17N'– 4#"#'"/#"'p=%"'&160%'1=)'G#2'R'"/#"'gp=%"'-%Eh'– O0';1'71"'#:G#2%'Y71G'/1G'-"'G#%'1A"#-70;E' P. Nahas
UC'© Joseph J. Nahas 2012
10 Dec 2012
,#%-&'40P7-$17%'?&17"E@'• >0#%=)0607"N'
– ./0'#%%-H7607"'18'7=60)#:%'"1'1Ap0&"%'#7;'0K07"%'#&&1);-7H'"1')=:0%E'
• 4#"#N'– <#7'A0'7=60)-&#:'1)'"0V"=#:'-7'81)6E'
– <#7'A0'%1=)&0%'18'-781)6#$17E'
In God we trust, all others bring data! Stu Hunter
P. Nahas
UU'© Joseph J. Nahas 2012
10 Dec 2012
()1A#A-:-"2'4-%")-A=$17'M=7&$17'• ()1A#A-:-"2'4-%")-A=$17'M=7&$17%N'
– 40%&)-A0;'A2'"/0'5)1A#A-:-"2';07%-"2'8=7&$17'8?V@'G/0)0'
G/0)0'\'-%'"/0')#7H0'18'VE'!
f (x)dx = 1R"
!
f (x) " 0,x# R
!
P(x" A) = f (x)dxA#
!
P(x = b) = f (s)dx =b
b
" 0
P. Nahas
A
NIST ESH 1.3.6 U3'© Joseph J. Nahas 2012
10 Dec 2012
<=6=:#$K0'4-%")-A=$17'M=7&$17'• <=6=:#$K0'4-%")-A=$17'M=7&$17'?<4M@'M?V@'G/0)0'
!
limx"#$
F(x) = 0
!
limx"+#
F(x) =1
!
f (x) = " F (x)
P. Nahas
NIST ESH 1.3.6.2
UB'© Joseph J. Nahas 2012
10 Dec 2012
9'60#%=)0'18'Z1&#$17N'>0#7'• >0#7'
– 9:%1'Y71G7'#%'./0'M-)%"'>1607"E'
– ./0'60#7'18'#'%=6'q'"/0'%=6'18'"/0'60#7%E'
– ./0'60#7'-%'#':-70#)'8=7&$17E'
– ./0'0%$6#"0'18'"/0'60#7'?%#65:0'#K0)#H0@N'
G/0)0'7'-%'"/0'7=6A0)'18'-"06%'-7'"/0'%#65:0E'
!
µ = E(X) = xf (x)x"R# dx
!
µx+y = µx + µy
!
µa+bx = a + bµx
P. Nahas
!
x =
xii=1
n
"n
NIST ESH 1.3.5.1
!
µ = E(X) = xf (x)x"R#
UD'© Joseph J. Nahas 2012
10 Dec 2012
a"/0)'>0#%=)0%'18'Z1&#$17'• >0;-#7'
– ./0'6-;51-7"'18'"/0';-%")-A=$17E'! ./0)0'#)0'#%'6#72'51-7"'#A1K0'#7;'A0:1G'"/0'60;-#7E'
• >1;0'– ./0'50#Y'K#:=0'18'"/0';-%")-A=$17E'
U`'© Joseph J. Nahas 2012
10 Dec 2012
>0#%=)0%'18'!5)0#;'• ]#)-#7&0'
– 9:%1'Y71G7'#%'./0'!0&17;'>1607"E'
– *8'#7;'17:2'-8'V'#7;'2'#)0'-7;0507;07"N'
– %3'-%'"/0'K#)-#7&0'0%$6#"0'?%#65:0'K#)-#7&0@N'
G/0)0'7'-%'"/0'7=6A0)'18'-"06%'-7'"/0'%#65:0E!
• !"#7;#);'40K-#$17'– "'-%'"/0'%"#7;#);';0K-#$17E'– %'-%'"/0'%"#7;#);';0K-#$17'0%$6#"0'?%#65:0'%"#7;#);';0K-#$17@E'
!
" 2 = E[(X # µ)2] = (x # µ)2 f (x)x$ f% dx
!
" x+y2 =" x#y
2 =" x2 +" y
2
P. Nahas !
s2 =
(xii=1
n
" # x)2
n #1
NIST ESH 1.3.5.6 UI'© Joseph J. Nahas 2012
10 Dec 2012
<1))0:#$17'Z0"'r'#7;'s-F'-q'U'"1'7'A0')#7;16'K#)-#A:0%F'#7;'
G/0)0'µ-'-%'"/0'60#7'18's-'#7;'"-3'-%'"/0'K#)-#7&0'18's-'
"/07'"/0'K#)-#7&0'18'2'
G/0)0'#-p'-%'"/0'&1))0:#$17'&10^&-07"'81)'"/0'515=:#$17's-spE'
!
Y = aii=1
n
" Xi
!
= " y2
= E[(Y # µ y )2]
= E[ (aii=1
n
$ xi # aiµi )2]
= ai2
i=1
n
$ " i2 + 2 ai
j$
i< j$ a j% ij" i" j
P. Nahas
Uj'© Joseph J. Nahas 2012
10 Dec 2012
./0'<1))0:#$17'<10^&-07"'
./0'&1))0:#$17'&10^&-07"'#, -%'#'%"#$%$&#:'61607"'"/#"'H-K0%'#' 60#%=)0' 18' !"#$%&' ($)$#($#*$' A0"G007' "G1' )#7;16'K#)-#A:0%E'*"'-%'$+,-%.$('A2N'
where sx and sy are the square roots of the estimates of the variance of x and y, while sxy is an estimate of the covariance of the two variables and is estimated by:
Poolla & Spanos
!
r =sxysxsy
!
sxy =
(xii=1
n
" # x)(yi # y)
n # 1
Uk'© Joseph J. Nahas 2012
10 Dec 2012
<1))0:#$17'• *8'_'q'UF'"G1')0#7;16'K#)-#A:0%'#)0'&1))0:#"0;E'
• *8'_''q'CF'"G1')#7;16'K#)-#A:0%'#)0'71"'&1))0:#"0;E'• *8'_''q'TUF'"G1')#7;16'K#)-#A:0%'#)0'-7K0)%0:2'&1))0:#"0;E'
• WV#65:0N'– ./0'/0-H/"'#7;'G0-H/"'18'#'%#65:0'18'"/0'515=:#$17'18'5015:0E'
! r1='G1=:;'0V50&"'#'51%-$K0'&1))0:#$17E''
Un'© Joseph J. Nahas 2012
10 Dec 2012
<1))0:#$17'WV#65:0'
Poolla & Spanos
3C'© Joseph J. Nahas 2012
10 Dec 2012
<1))0:#$17'WV#65:0'
Poolla & Spanos
NIST ESH 3.4.2.1
3U'© Joseph J. Nahas 2012
10 Dec 2012
!"#$%$&'a=":-70'UE ,#&YH)1=7;N'
9E O/2'!"=;2'!"#$%$&%'#7;'!"#$%$&#:'WV50)-607"#:'40%-H7b',E \080)07&0%'
3E ,#%-&'!"#$%$&#:'./01)2'9E ,#%-&'!"#$%$&#:'40P7-$17%'
-E 4-%")-A=$17%'--E !"#$%$&#:'>0#%=)0%'---E *7;0507;07&0c40507;07&0'
#E <1))0:#$17'<10^&-07"'AE <1))0:#$17'WV#65:0'&E <1))0:#$17'<10^&-07"'#7;']#)-#7&0'
,E ,#%-&'4-%")-A=$17%'-E 4-%&)0"0'K%E'<17$7=1=%'4-%")-A=$17%'--E ,-716-#:'4-%")-A=$17'---E L1)6#:'4-%")-A=$17'-KE ./0'<07")#:'Z-6-"'./01)06'
#E 40P7-$17'AE 4-&0'#%'#7'0V#65:0'
33'© Joseph J. Nahas 2012
10 Dec 2012
./0'4-%")-A=$17'18'V'
• <#7'&#:&=:#"0'"/0'5)1A#A-:-"2'18'#')#7;16:2'&/1%07'1A%0)K#$17'18'"/0'515=:#$17'8#::-7H'G-"/-7'#'H-K07')#7H0'R'%1'-"'-%'#'5)1A#A-:-"2';-%")-A=$17E'
• ]0)$&#:'1);-7#"0F'(?V@'-%'&#::0;'"/0'5)1A#A-:-"2';07%-"2'• <#7'G0'P7;'#'6#"/06#$&#:'8=7&$17'"1';0%&)-A0'"/0'
5)1A#A-:-"2';-%")-A=$17b'
Discrete Distributions Continuous Distributions Shape = f(x)
Center = mean Spread = variance
Poolla & Spanos
NIST ESH 1.3.6.6
3B'© Joseph J. Nahas 2012
10 Dec 2012
9'4-%&)0"0'4-%")-A=$17'
What discrete distribution is this?
3D'© Joseph J. Nahas 2012
10 Dec 2012
9'4-%&)0"0'4-%")-A=$17N'"/0',-716-#:'• ./0'A-716-#:';-%")-A=$17'-%'=%0;'G/07'"/0)0'#)0'0V#&":2'"G1'
6="=#::2'0V&:=%-K0'1="&160%'18'#'")-#:E''– ./0%0'1="&160%'#)0'#55)15)-#"0:2':#A0:0;'t%=&&0%%t'#7;'t8#-:=)0tE''
• ./0'A-716-#:';-%")-A=$17'-%'=%0;'"1'1A"#-7'"/0'5)1A#A-:-"2'18'1A%0)K-7H'V'%=&&0%%0%'-7'7'")-#:%F'G-"/'"/0'5)1A#A-:-"2'18'%=&&0%%'17'#'%-7H:0'")-#:';071"0;'A2'5E''– ./0'A-716-#:';-%")-A=$17'#%%=60%'"/#"'5'-%'PV0;'81)'#::'")-#:%E'
NIST ESH 1.3.6.6.18
where:
Mean = np Standard Deviation =
3`'© Joseph J. Nahas 2012
10 Dec 2012
9'4-%&)0"0'4-%")-A=$17N'"/0',-716-#:'
• !-65:0'WV#65:0%N'– L=6A0)'18'/0#;%'-7'UC'&1-7'u-5%N'5'q'CE`F'7'q'UC'
– L=6A0)'18'170%'-7'`')1::%'18'#';-0N'5'q'UcIF'7'q'`'
Binomial Distribution with p = 0.10 and n=15.
NIST ESH 1.3.6.6.18 3I'© Joseph J. Nahas 2012
10 Dec 2012
WV#65:0N'v%-7H'"/0',-716-#:'4-%")-A=$17'• ./0'5)1A#A-:-"2'"/#"'#'6061)2'&0::'8#-:%'-%'UCTnE''
• *7'#'ID'>A-"'6061)2'#))#2'G/#"'-%'"/0''5)1A#A-:-"2'"/#"N'– 9::'&0::%'#)0'awb'– U'&0::'8#-:0;b'– `'&0::%'8#-:0;b'– >1)0'"/#7'`'&0::%'8#-:0;b'
Poolla & Spanos
3j'© Joseph J. Nahas 2012
10 Dec 2012
,-716-#:'!1:=$17'
!
P(X = 0) =nx"
# $ %
& ' px (1( p)n(x
=n0"
# $ %
& ' (10(9)0(1( (10(9))n(0
=1•1• (1( (10(9))n
)1( n(10(9) + * * *
)1( 67•106 •10(9
) 0.933
64 Mb Memory = n = 226 = 6.7E+7
3k'© Joseph J. Nahas 2012
10 Dec 2012
9'<17$7=1=%'4-%")-A=$17N'"/0'L1)6#:'
figure 2-10 Montgomery pp 39
- $ < x < $"
NIST ESH 1.3.6.6.1
Also called a Gaussian Distribution
Notation: x~N(μ,σ)$i.e.: x is Normally Distributed with a mean of μ$
and a standard deviation of σ. $
3n'© Joseph J. Nahas 2012
10 Dec 2012
!"#7;#);'L1)6#:'4-%")-A=$17'
-%'"/0'!"#7;#);'L1)6#:'4-%")-A=$17N'
%1'-8 ' '''F'"/07'
./0)0'#)0'"#A:0%'18'
!
"(z) =0
z
# e $%2 / 2
2&d%!
z =x " µ#
~ N(0,1)!
z ~ N(0,1)
!
x ~ N(µ," )
P. Nahas
i.e. µ = 0, ! = 1
NIST ESH 1.3.6.7.1
!
f (z) =e " x
2 / 2
2#
BC'© Joseph J. Nahas 2012
10 Dec 2012
L1)6#:'4-%")-A=$17'.#A:0'8)16'W!d'
NIST ESH 1.3.6.7.1
f(z) = f(-z) Note: Area from 0 to +" = 0.5
BU'© Joseph J. Nahas 2012
10 Dec 2012
WV#65:0N'.#A:0'Z11Y=5'81)'L1)6#:'4-%")-A=$17'• ./0'G#80)T"1TG#80)'"/-&Y70%%'18'#'51:2':#20)'-%';-%")-A="0;'
71)6#::2'#)1=7;'`CC76'G-"/'#'"'18'3C76N'– '''''''''''''''("/'x'L'?`CC'76F'3C'76@'
• O/#"'-%'"/0'5)1A#A-:-"2'"/#"'#'H-K07'G#80)'G-::'/#K0'51:2%-:-&17'"/-&Y0)'"/#7'`UC76b'
• EEE'"/-770)'"/#7'DkC76b'
• EEE'A0"G007'DnC'#7;'`U`76b'
Poolla & Spanos
NIST ESH 1.3.6.7.1 B3'© Joseph J. Nahas 2012
10 Dec 2012
WV#65:0N'.#A:0'Z11Y=5'81)'L1)6#:'4-%")-A=$17'• ./0'G#80)T"1TG#80)'"/-&Y70%%'18'#'51:2':#20)'-%';-%")-A="0;'
71)6#::2'#)1=7;'`CC76'G-"/'#'"'18'3C76N'– '''''''''''''''("/'x'L'?`CC'76F'3C'76@'
• O/#"'-%'"/0'5)1A#A-:-"2'"/#"'#'H-K07'G#80)'G-::'/#K0'51:2%-:-&17'"/-&Y0)'"/#7'`UC76b'– `UC'R'`CC'76'q'UC'76'q'CE`'"'8)16'60#7'– M)16'"#A:0'C'"1'CE`'"'q'CEUn'81)'A0"G007'`CC'#7;'`UC'76E'– X)0#"0)'"/#7'`UC'76'q'CE`'R'CEUn'q''CEBU'
• EEE'"/-770)'"/#7'DkC76b'– `CC'R'DkC'76'q'3C'76'q'U'"'8)16'60#7'– M)16'"#A:0'C'"1'U'"'q'CEBD'81)'A0"G007'DkC'#7;'`CC'76'– ./-770)'"/#7'DkC'q'CE`'R'CEBD'q'CE3I'
• EEE'A0"G007'DnC'#7;'`U`76b'– `CC'R'DnC'76'q'CE`'"'8)16'60#7y'`U`'R'`CC'q'CEj`'"'8)16'60#7'– M)16'"#A:0N'CEUn'z'CE3j'q'CEDI'
Poolla & Spanos
NIST ESH 1.3.6.7.1
BB'© Joseph J. Nahas 2012
10 Dec 2012
L1)6#:'4-%")-A=$17'.#A:0'8)16'W!d'
NIST ESH 1.3.6.7.1 BD'© Joseph J. Nahas 2012
10 Dec 2012
./0'9;;-$K-"2'18']#)-#7&0'• *M'2'q'#UVUz'#3V3'z'{z#7V7''
– "/07'Q2'q'#UQUz'#3Q3'z'{z#7Q7''
– #7;'S23'q'#U3SU3'z'#33S33'z'{z'#73S73''– ./-%'#55:-0%'#55:-0%'=7;0)'"/0'#%%=65$17'"/#"'"/0'5#)#60"0)%'V-'#)0'
"#($)$#($#.EE'
WV#65:0%N'
• ./0'"/-&Y70%%'K#)-#7&0'18'#':#20)';0P70;'A2'"G1'&17%0&=$K0'H)1G"/%N'– Q"'q'QHU'z'QH3''
– S"3'q'SHU3'z'SH33''• ./0'"/-&Y70%%'K#)-#7&0'18'#'H)1G"/'%"05'81::1G0;'A2'#7'0"&/'
%"05N'– Q"'q'QH'+'Q0''
– S"3'q'SH3'z'S03''
B`'© Joseph J. Nahas 2012
10 Dec 2012
WV#65:0N'd1G'"1'<16A-70'<17%0&=$K0'!"05%'• ./0'"/-&Y70%%'18'#'!-a3':#20)'-%';-%")-A="0;'71)6#::2'#)1=7;'
ICC76'G-"/'#'"'18'3C76N'– X1V'x'L'?ICC76F'3C76@'
• 4=)-7H'#'51:2%-:-&17')061K#:'%"05'G-"/':-6-"0;'%0:0&$K-"2F'%160'18'"/0'1V-;0'-%')061K0;E'./0')061K0;'1V-;0'-%N'– \1V'xL'?`C76F'`76@'
• O/#"'-%'"/0'5)1A#A-:-"2'"/#"'"/0'P7#:'1V-;0'"/-&Y70%%'-%'A0"G007'`DC'#7;'`IC76b'
Poolla & Spanos
BI'© Joseph J. Nahas 2012
10 Dec 2012
WV#65:0N'd1G'"1'<16A-70'<17%0&=$K0'!"05%'• ./0'"/-&Y70%%'18'#'!-a3':#20)'-%';-%")-A="0;'71)6#::2'#)1=7;'
ICC76'G-"/'#'"'18'3C76N'– X1V'x'L'?ICC'76F'3C'76@'
• 4=)-7H'#'51:2%-:-&17')061K#:'%"05'G-"/':-6-"0;'%0:0&$K-"2F'%160'18'"/0'1V-;0'-%')061K0;E'./0')061K0;'1V-;0'-%N'– \1V'xL'?`C76F'`76@'
• O/#"'-%'"/0'5)1A#A-:-"2'"/#"'"/0'P7#:'1V-;0'"/-&Y70%%'-%'A0"G007'`DC'#7;'`IC76b'
• <#:&=:#$17%N'– QW'q'QX'R'Q\'q'ICC'R'`C'76'q'``C'76'
– "W3'q'"X
3'z'"\3'q'3C3'z'`3'763'q'D3`'763y'"W'q'3CEI'76'
– W1V'xL?``C'76F'3CEB'76@'
– `DC'76'q'QW'R'CEDn'"Wy'`IC'q'QW'z'CEDn'"W''
– CEUn'z'CEUn'q'CEBk' Poolla & Spanos
Bj'© Joseph J. Nahas 2012
10 Dec 2012
./0'<07")#:'Z-6-"'./01)06N''• ./0';-%")-A=$17'18'#'%=6'1)'#K0)#H0'18'6#72')#7;16'K#)-#A:0%'-%'&:1%0'"1'
71)6#:E'– ./-%'-%'")=0'0K07'-8'"/0'K#)-#A:0'#)0'71"'-7;0507;07"'#7;'0K07'-8'"/02'/#K0'
;-m0)07"';-%")-A=$17%E'
• >1)0'1A%0)K#$17%'#)0'700;0;'-8'"/0';-%")-A=$17'%/#50'-%'8#)'8)16'71)6#:E'• L1';-%")-A=$17'%/1=:;'A0';16-7#7"E'
Sum of 5 unif. distr. numbers: Uniformly distributed number
P Nahas and Poolla & Spanos
NIST ESH 1.3.6.6.1 Bk'© Joseph J. Nahas 2012 10 Dec 2012
4-&0'#7;'"/0'<07")#:'Z-6-"'./01)06'
5(675'
5(685'
5(695'
5(6:5'
5(;55'
6' ;' <' =' >' 7'
()1A#A-:-"2'!-7H:0'4-0'
5(555'
5(5>5'
5(655'
5(6>5'
5(;55'
;' =' 7' 9' 65' 6;'
()1A#A-:-"2'.G1'4-0'
5(555'
5(5>5'
5(655'
5(6>5'
5(;55'
<' =' >' 7' 8' 9' :' 65'66'6;'6<'6='6>'67'68'69'
()1A#A-:-"2'./)00'4-0'