13 s241 functions of random variables
TRANSCRIPT
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Methods for determining the distribution of
functions of Random Variables
1. Distribution function method
2. Moment generating function method
3. Transformation method
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Distribution function method
Let X, , ! ". ha#e $oint densit%f&x,y,z, '
Let W ( h&X, Y, Z, '
First stepFind the distribution function of W
G&w' (P)W * w+ (P)h&X, Y, Z, '* w+
Second stepFind the densit% function of W
g&w' ( G'&w'.
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-amle 1
Let X ha#e a normal distribution /ith mean 0, and
#ariance 1. &standard normal distribution'
Let W (X2.
Find the distribution of W.
( )
2
21
2
x
f x e
=
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First step
Find the distribution function of WG&w' (P)W * w+ (P)X2* w+
if 0P w X w w
= 2
21
2
w x
w
e dx
=
( ) ( )F w F w=
/here ( ) ( )
2
21
2
x
F x f x e
= =
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( ) ( ) ( )d wd w
F w F wdw dw
=
Second step
Find the densit% function of W
g&w' ( G'&w'.
1 1
2 2 2 2
1 1 1 1
2 22 2
w w
e w e w
= +
( ) ( )1 1
2 21 12 2
f w w f w w = +
1
2 21
if 0.
2
w
w e w
=
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Thus ifX has a standard ormal distribution then
W = X2
has densit%
( )1
2 21
if 0.
2
w
g w w e w
=
This distribution is the amma distribution /ith =
and ( .
This distribution is also the2distribution /ith = 1
degree of freedom.
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-amle 2
4uose thatXand Y are indeendent random#ariables each ha#ing an e-onential distribution
/ith arameter &mean 15'
Let W (X6 Y.
Find the distribution of W.
( )1 for 0x
f x e x
= ( )2 for 0
yf y e y =
( ) ( ) ( )1 2,f x y f x f y=( )2 for 0, 0x y
e x y
+=
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First step
Find the distribution function of W = X 6 YG&w' (P)W * w+ (P)X6 Y * w+
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[ ] ( ) ( )1 20 0
w w x
P X Y w f x f y dydx
+ =
( )2
0 0
w w xx y
e dydx
+=
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[ ] ( ) ( )1 20 0
w w x
P X Y w f x f y dydx
+ =
( )2
0 0
w w xx y
e dydx
+=
2
0 0
w w xx ye e dy dx =
2
0 0
w xw yx e
e dx
= ( ) 0
2
0
w w x
x e ee dx
=
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[ ]P X Y w+ ( ) 0
2
0
w w x
x e ee dx
=
0
w
x we e dx =
0
wx
we xe
=
0wwe ewe
=
1 w we we =
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Second step
Find the densit% function of W
g&w' ( G'&w'.
1 w wd
e wedw
=
ww wdw dee e w
dw dw
= +
2w w we e we = +
2 for 0wwe w =
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7ence ifX and Y are indeendent random #ariableseach ha#ing an e-onential distribution /ith arameter
then W has densit%
( ) 2 for 0wg w we w =
This distribution can be recogni8ed to be the Gammadistribution /ith arameters = 2 and .
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-amle9 4tudent:s t distribution
LetZ and Ube t/o indeendent random#ariables /ith9
1. Z ha#ing a 4tandard ormal distribution
and
2. U ha#ing a2distribution /ith degreesof freedom
Find the distribution ofZ
tU
=
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The densit% ofZ is9
( )
2
2
1
2
z
f z e
=
The densit% of U is9
( )
2
12 2
12
2
u
h u u e
=
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Therefore the $oint densit% ofZ and Uis9
The distribution function of Tis9
( ) [ ]Z t
G t P T t P t P Z U U
= = =
( ) ( ) ( )
2
2
12 2
1
2,
2 2
z u
f z u f z h u u e
+
= =
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Therefore9
( ) [ ] tG t P T t P Z U = = =
22
12 2
0
12
2
2
t uz u
u e dzdu
+
=
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;llustration of limits
U
U
zz
t > 0 t > 0
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o/9
22
12 2
0
12
& '
22
t uz u
G t u e dzdu
+
=
and9
( )
2
2
12 2
0
1
2& '
2
2
t
u z ud
g t G t u e dz dudt
+
= =
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7ence
221
1
2 2
0
12
& '
2
2
t u
g t u e du
+ =
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7ence2 1
1 21
2 21
20 2
12
2
1
tu
u e du
t
+ +
+
+ =
+
and1
212
2 2
1 12
2 2& ' 1
22
tg t
++
+
= +
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or
1 12 22 2
12
& ' 1 1
2
t tg t K
+ + + = + = +
1
2
2
K
+
=
/here
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Students t distribution
12 2
& ' 1t
g t K
+
= +
1
2
2
K
+
=
/here
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4tudent = >.>. osset
>or?ed for a distiller%
ot allo/ed to ublish
@ublished under the
seudon%m A4tudent
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t distribution
standard normal distribution
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Distribution of the Ma- and Min
4tatistics
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Letx1,x2, " ,xdenote a samle of si8e from
the densit%f&x'.
Let!( ma-&x"' then determine the distribution
of!.
Reeat this comutation for # ( min&x"'
Bssume that the densit% is the uniform densit%
from 0 to .
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7ence
10& '
else/here
xf x
=
0
and the distribution function
[ ]
0 0
& ' 0
1
x
xF x P X x x
x
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Finding the distribution function of!.
[ ] ( )& ' ma- "G t P ! t P x t = =
[ ]1 , , P x t x t= L
[ ] [ ]1 P x t P x t= L
0 0
0
1
t
tt
t
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Differentiating /e find the densit% function of!.
( ) ( )
1
0
0 other/ise
tt
g t G t
= =
f&x' g&t'
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Finding the distribution function of #.
[ ] ( )& ' min "G t P # t P x t = =
[ ]11 , , P x t x t= > >L
[ ] [ ]11 P x t P x t= > >L
0 0
1 1 0
1
t
tt
t
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Differentiating /e find the densit% function of #.
( ) ( )
1
1 0
0 other/ise
t
tg t G t
= =
f&x' g&t'
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The robabilit% integral transformation
This transformation allo/s one to con#ert
obser#ations that come from a uniform
distribution from 0 to 1 to obser#ations that
come from an arbitrar% distribution.
Let U denote an obser#ation ha#ing a uniform
distribution from 0 to 1.
1 0 1& '
else/here
ug u
=0
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Find the distribution ofX.
1& 'X F U=Let
Letf&x$denote an arbitrar% densit% function and
F%x' its corresonding cumulati#e distribution
function.
( ) [ ] 1& 'G x P X x P F U x = =
( )P U F x =
( )F x=7ence.
( ) ( ) ( ) ( )g x G x F x f x = = =
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The Transformation Method
TheoremLet X denote a random #ariable /ith
robabilit% densit% functionf&x' and U (
h&X'.
Bssume that h&x' is either strictl% increasing
&or decreasing' then the robabilit% densit% of
U is9
( ) ( ) ( )1
1 & '& ' dh u dx
g u f h u f xdu du
= =
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Proof
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( )
( )
1
1
& ' strictl% increasing
1 & ' strictl% decreasing
F h u h
F h u h
=
hence
( ) ( )g u G u=
( )( ) ( )
( )( ) ( )
1
1
1
1
strictl% increasing
strictl% decreasing
dh u
F h u hdu
dh uF h u h
du
=
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or
( ) ( ) ( )1
1 & '& ' dh u dx
g u f h u f xdu du
= =
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-amle
4uose that X has a ormal distribution/ith meanand #ariance 2.
Find the distribution of U ( h&x' ( eX.
Solution:
( )( )
2
221
2
x
f x e
=
( ) ( ) ( ) ( )11 ln 1ln and
dh u d uh u u
du du u
= = =
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hence
( ) ( ) ( )1
1 & '& ' dh u dx
g u f h u f xdu du
= =
( )( )2
2
ln
21 1
for 02
u
e uu
= >
This distribution is called the logCnormal
distribution
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logCnormal distribution
The Transfomation Method
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The Transfomation Method
&man% #ariables'Theorem
Let x1,x2,",xdenote random #ariables /ith
$oint robabilit% densit% function
f&x1,x2,",x'
Let u1( h1&-1,x2,",x'.
u2( h2&-1,x2,",x'.
u( h&-1,x2,",x'.
define an in#ertible transformation from thex:s to the u:s
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Then the $oint robabilit% densit% function of
u1, u2,",uis gi#en b%9
( ) ( ) ( )
( )1
1 1
1
, ,, , , ,
, ,
d x xg u u f x x
d u u=
LL L
L
( )1, , f x x &= L
/here( )
( )
1
1
, ,
, ,
d x x&
d u u=
L
L
acobian of the transformation
1 1
1
1
det
dx dx
du du
dx dx
du du
=
L
M M
K
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-amle4uose that x1,x2 are indeendent /ith densit%
functionsf1 &x1' andf2&x2'
Find the distribution of
u1(x16x2u2(x1Cx2
4ol#ing forx1and x /e get the in#erse transformation
1 21
2u ux +=
1 22
2
u ux
=
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( )
( )1 2
1 2
,
,
d x x&
d u u=
The acobian of the transformation
1 1
1 2
2 2
1 2
det
dx dxdu du
dx dx
du du
=
1 1
1 1 1 1 12 2det 1 1 2 2 2 2 2
2 2
= = =
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The $oint densit% ofx1,x2 is
f&x1,x2' (f1 &x1'f2&x2'
7ence the $oint densit% of u1and u2is9
1 2 1 21 2
1
2 2 2
u u u uf f
+ =
( ) ( )1 2 1 2, ,g u u f x x &=
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From
( )
1 2 1 2
1 2 1 2
1,
2 2 2
u u u ug u u f f
+
=
>e can determine the distribution of u1(x16x2
( ) ( )1 1 1 2 2,g u g u u du
=
1 2 1 21 2 2
1
2 2 2
u u u uf f du
+ =
1 2 1 21
2
1ut then ,
2 2 2
u u u u d(( u (
du
+ = = =
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7ence
( ) 1 2 1 21 1 1 2 21
2 2 2
u u u ug u f f du
+ =
( ) ( )1 2 1f ( f u ( d(
=
This is called the con#olution of the t/o
densitiesf1andf2.
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Example: The e-Caussian distribution
1. X has an e-onential distribution /ith
arameter .
2. Y has a normal &aussian' distribution /ith
meanand standard de#iation .
LetX and Ybe t/o indeendent random
#ariables such that9
Find the distribution of U (X 6 Y.
This distribution is used in s%cholog% as a
model for resonse time to erform a tas?.
0x
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o/ ( )10
0 0
xe xf x
x
=
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or
( )( )
2
22
02
u ((
g u e d(
=
( )
2 2
2
2
2
02
u ( (
e d(
+ =
( ) ( )22 2
2
2 2
2
02
( u ( u (
e d(
+ + =
( ) ( )2 22
2 2
2
2 2
02
( u (u
e e d(
=
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or ( ) ( )2 22
2 2
2
2 2
02
( u (u
e e d(
=
( ) ( ) ( ) ( )2 22 2 2 2 2
2 2
2
2 2
02
u u ( u ( u
e e d(
+ =
( ) ( ) ( ) ( )
2 22 2 2 2 2
2 2
2
2 2
0
1
2
u u ( u ( u
e e d(
+ =
( ) ( )
[ ]
22 2
22 0
u u
e P +
=
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>here + has a ormal distribution /ith mean
( )( )
22
2
21
u ug u e
+ =
( )2
+ u = +and #ariance 2.
7ence
>here &z' is the cdf of the standard ormaldistribution
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g&u'
The e-Caussian distribution
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The distribution of a random #ariableX is described b%
either
1. The densit% functionf&x' ifX continuous &robabilit% mass function&x' if
X discrete', or
2. The cumulati#e distribution functionF&x', or
3. The moment generating function #X&t'
P ti
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Properties
1. #X&0' ( 1
( ) ( ) ( )0 deri#ati#e of at 0. th
X X# . # t t = =2.
( )
.
. - X= =
( ) 2 33211 .2E 3E E
X# t t t t t
.
= + + + + + +L L3.
( ) ( )
( )
continuous
discrete
.
.
. .
x f x dx X- X
x , x X
= =
Let X be a random ariable ith moment
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. LetXbe a random #ariable /ith moment
generating function #X&t'. Let Y( bX 6 a
Then #Y&t' ( #bX 6 a&t'(-&e )bX 6 a+t' ( eat-&e X) bt +'
( eat#X &bt'
G. LetXand Ybe t/o indeendent random
#ariables /ith moment generating function
#X&t' and #Y&t' .
Then #X/Y&t' (-&e)X 6 Y+t' (-&e Xt e Yt'
(-&e Xt'-&e Yt'
( #X &t' #Y &t'
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H. LetXand Ybe t/o random #ariables /ith
moment generating function #X&t' and #Y&t'
and t/o distribution functionsFX&x' and
FY&y' resecti#el%.
Let #X &t' ( #Y &t' thenFX&x' (FY&x'.
This ensures that the distribution of a random
#ariable can be identified b% its moment
generating function
M F : I ti di t ib ti
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M. . F.:s C Iontinuous distributions
M F : Di t di t ib ti
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M. . F.:s C Discrete distributions
Moment generating function of the
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Moment generating function of the
gamma distribution
( ) ( ) ( )tX txX# t - e e f x dx
= =
( ) ( )1 0
0 0
xx e xf x
x
=
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( ) ( ) ( )tX txX# t - e e f x dx
= =
( )1
0
tx xe x e dx
=
using
( )( )1
0
t xx e dx
=
( )1
0
1
a
a bxb
x e dxa
=( )1
0
a bx
a
ax e dx
b
=
or
then
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then
( ) ( )( )1
0
t x
X# t x e dx
=
( )
( )
( )t
=
tt
= <
Moment generating function of the
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Moment generating function of the
4tandard ormal distribution
( ) ( ) ( )tX txX# t - e e f x dx
= =
( )
2
21
2
x
f x e
=
/here
thus
( )
2 2
2 21 1
2 2
x xtx
tx
X# t e e dx e dx
+
= =
( )2
1x a
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>e /ill use( )
22
0
11
2
x a
be dxb
=
( )
2
21
2
xtx
X# t e dx
+
= 2
221
2
x tx
e dx
= ( )
22 2 2 22
2 2 2 2
1 1
2 2
x tx tx t t t
e e dx e e dx
+
= =
2
2
t
e= 2 3
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ote9
( )
2
2 32 2
2
22 2
12 2E 3E
t
X
t t
t# t e
= = + + + +L
2 3
12E 3E FE
x x x xe x= + + + + +L
2 H 2
2 31
2 2 2E 2 3E 2 E
#
#
t t t t
#= + + + + + +L L
Blso
( ) 2 332112E 3E
X# t t t t
= + + + +L
2 3 x x x
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ote9
( )
2
2 32 2
2
22 2
12 2E 3E
t
X
t t
t# t e
= = + + + +L
12E 3E FE
x x x xe x= + + + + +L
2 H 2
2 31
2 2 2E 2 3E 2 E
#
#
t t t t
#= + + + + + +L L
Blso ( ) 2 332112E 3E
X# t t t t = + + + +L
( )momentth x f x dx
= =
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Juating coefficients of t, /e get
( )
21for 2 then2 E 2 E
#
#. #
# #
= =
0 if is odd and . =
1 2 3 hence 0, 1, 0, 3 = = = =
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Thus Y = aX / b has a normal distribution /ith
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ThusZ has a standard normal distribution .
Special Case:thez transformation
1XZ X aX b
= = + = +
10Z a b
= + = + =
2
2 2 2 21 1Z a
= = =
ThusY = aX / b has a normal distribution /ith
mean a/ band #ariance a22.
-amle
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-amle4uose thatXand Yare indeendent eachha#ing a
normal distribution /ith meansX andY , standardde#iations X and Y
Find the distribution of = X / Y
( )
2 2
2XX
tt
X# t e
+=
Solution:
( )
2 2
2
YY
tt
Y# t e
+=
( ) ( ) ( )
2 2 2 2
2 2
X YX Y
t tt t
X Y X Y# t # t # t e e
+ +
+ = =
o/
or
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or
( ) ( )
( )2 2 2
2
X Y
X Y
tt
X Y# t e
++ +
+ =
( the moment generating function of thenormal distribution /ith meanX /Yand
#ariance2 2
X Y +
ThusY = X / Y has a normal distribution
/ith meanX /Yand #ariance2 2
X Y
+
-amle
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-amle4uose thatXand Yare indeendent eachha#ing anormal distribution /ith means
Xand
Y, standard
de#iations X and Y
Find the distribution of1 = aX / bY
( )
2 2
2XX
tt
X# t e
+=
Solution:
( )
2 2
2
Y
Y
t
tY# t e
+=
( ) ( ) ( ) ( ) ( )aX bY aX bY X Y # t # t # t # at # bt + = =
o/
( ) ( )
( ) ( )
2 22 2
2 2
X YX Y
at bt at bt
e e
+ +
=
or
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or
( ) ( )
( )2 2 2 2 2
2
X Y
X Y
a b ta b t
aX bY # t e
++ +
+ =
( the moment generating function of thenormal distribution /ith mean aX / bY
and #ariance2 2 2 2
X Ya b +
ThusY = aX / bY has a normal
distribution /ith mean aX / bYand
#ariance2 2 2 2
X Ya b +
4ecial Iase9
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4ecial Iase9
ThusY = X 2 Y has a normal distribution
/ith meanX 2Yand #ariance
( ) ( )2 22 2 2 2
1 1X Y X Y
+ + = +
a ( 61 and b( C1.
-amle &-tension to indeendent RV:s'
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-amle &-tension to indeendent RV s'4uose thatX1,X2, ",Xare indeendent eachha#ing a
normal distribution /ith means", standard de#iations "&for " = 1, 2, " ,'
Find the distribution of1 = a1X1/ a1X2/ / aX
( )
2 2
2""
"
tt
X# t e
+=
Solution:
( ) ( ) ( )1 1 1 1 a X a X a X a X
# t # t # t + + =L Lo/
( ) ( )
( ) ( )
22 221 1
1 12 2
a ta ta t a t
e e
+ +
= L
&for " = 1, 2, " ,'
( ) ( )1 1 X X
# a t # a t = L
or
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or
( ) ( )
( )2 2 2 2 21 11 1
1 1
......
2
a a ta a t
a X a X # t e
+ ++ + +
+ + =L
( the moment generating function of thenormal distribution /ith mean
and #ariance
ThusY = a1X1/ / aXhas a normal
distribution /ith mean a1
1
/ / a
and
#ariance
1 1 ... a a + +2 2 2 2
1 1
...
a a + +
2 2 2 21 1 ... a a + +
Special case:
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1 2
1a a a
= = = =L
1 2 = = = =L2 2 2 2
1 1 1 = = = =L
;n this caseX1,X2, ",Xis a samle from a
normal distribution /ith mean, and standardde#iations , and
( )1 21 1 X X X
= + + +L
the samle meanX= =
p
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Thus
2 2 2 2 21 1 ...x a a = + +
and #ariance
1 1 ...x a a = + +has a normal distribution /ith mean
1 1 ... Y x a x a x= = + +
( ) ( )11 1... x x = + +
( ) ( )1 1... = + + =
2 2 2 22 2 21 1 1...
= + + = =
Summary
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;fx1,x2, ",xis a samle from a normal
distribution /ith mean, and standardde#iations , then the samle meanx=
y
22
x
=
and #ariance
x =
has a normal distribution /ith mean
standard de#iation x
=
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@oulation
4amling distribution
ofx
The Law of Large umbers
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4uosex1,x2, ",xis a samle &indeendent
identicall% distributed = i.i.d.' from adistribution /ith mean,
the samle meanx=
g
Then
1 as for all 0P x < >
Let
Proof:@re#iousl% /e used Tcheb%che#:s Theorem.
This assumes &2' is finite.
Proof: &use moment generating functions'
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>e /ill use the follo/ing fact9
Let
#1&t', #2&t', "
denote a seJuence of moment generating functions
corresonding to the seJuence of distribution
functions9F1&x' ,F2&x', "
Let #&t' be a moment generating function
corresonding to the distribution functionF&x' then
if
& g g '
( ) ( )lim for all in an inter#al about 0.""
# t # t t
=
( ) ( )lim for all .""
F x F x x
=then
Let x x denote a seJuence of indeendent
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Letx1,x2, " denote a seJuence of indeendent
random #ariables coming from a distribution /ith
moment generating function #&t' and distributionfunctionF&x'.
( ) ( ) ( ) ( ) ( )1 2 1 2
( 0 x x x x x x
# t # t # t # t # t + + += L L
Let (x16x26 " 6xthen
( )(
# t
1 2no/ x x x x
+ + += =L
( ) ( )1or
x 00
t t# t # t # #
= = =
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using L:7oitals rule
( )no/ ln ln lnxt t
# t # #
= =
( )ln /here
t # u tu
u = =
( ) ( )0ln
Thus lim ln limx u
t # u# t
u = ( )
( ) ( )
( )00
lim 1 0u
# ut
# u #
t t#
= = =
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is the moment generating function of
a random #ariable that ta?es on the #alue/ithrobabilit% 1.
( ) ( )and lim for all #alues of .x
F x F x x
=
( ) t# t e=
( )1
i.e. andx
xx
==0
( )
0
and distribution function and1
x
F x x
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o/
( )0
since and
1
xF x
x
as
K..D.
The Central Limit theorem
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;fx1,x2, ",xis a samle from a distribution
/ith mean, and standard de#iations , thenif is large the samle meanx=
22
x
=
and #ariance
x =
has a normal distribution /ith mean
standard de#iation x
=
Proof: &use moment generating functions'
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>e /ill use the follo/ing fact9
Let
#1&t', #2&t', "
denote a seJuence of moment generating functions
corresonding to the seJuence of distribution
functions9F1&x' ,F2&x', "
Let #&t' be a moment generating function
corresonding to the distribution functionF&x' then
if ( ) ( )lim for all in an inter#al about 0.""
# t # t t
=
( ) ( )lim for all .""
F x F x x
=then
Let x x denote a seJuence of indeendent
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Letx1,x2, " denote a seJuence of indeendent
random #ariables coming from a distribution /ith
moment generating function #&t' and distributionfunctionF&x'.
( ) ( ) ( ) ( ) ( )1 2 1 2
( 0 x x x x x x
# t # t # t # t # t + + += L L
Let (x16x26 " 6xthen
( )(
# t
1 2no/ x x x x
+ + += =L
( ) ( )1or
x 00
t t# t # t # #
= = =
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Letx
z x
= =
( )then
t t
z x
t t # t e # e #
= =
( )and ln lnz t
# t t #
= +
2
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( )Then ln lnz t
# t t #
= +
( )2 2
2 2 2ln
t t# u
u u
= +
2
2 2Let or and
t t tu
u u = = =
( )2
2 2
ln # u ut
u
=
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( )( ) ( )( )0
o/ lim ln lim lnz z u
# t # t
=
( )2
2 20
lnlimu
# u ut
u
=
( )( )2
2 0lim using L7oitals rule
2u
# u
# ut
u
=
( ) ( ) ( )
( )
2
22
2 0lim using L7oitals rule again
2u
# u # u # u
# ut
=
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( ) ( ) ( )
( )
2
2
2
2 0lim using L7oitals rule again
2u
# u # u # u
# ut
=
( ) ( )
22
20 0
2# #t
=
( ) ( )222 2
2 2 2
" "- x - xt t
= =
( )( ) ( )( )2
2
2thus lim ln and lim2
t
z z
t# t # t e
= =
2t
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( )
2
2o/t
# t e=
;s the moment generating function of the standard
normal distribution
Thus the limiting distribution ofz is the standardnormal distribution
( )
2
21
i.e. lim 2
x u
z F x e du
=
Q.E.D.
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The Ientral Limit theorem
illustrated
The Central Limit theorem
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;fx1,x2, ",xis a samle from a distribution
/ith mean, and standard de#iations , thenif is large the samle meanx=
22
x
=
and #ariance
x =
has a normal distribution /ith mean
standard de#iation x
=
The Central Limit theorem illustrated
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;fx1,x2are indeendent from the uniform
distirbution from 0 to 1. Find the distributionof9 the samle meanx=
1 21 2 and 2 2
x x00 x x x += + = =
let
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0
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o/9 ( )122
0x 0 a0= = =
The densit% of is9x
( ) ( ) ( )2 2dh x g g xdx
= =
( )
12
12
2 0 2 1 2 0
2 2 1 2 2 2 1 1
0 other/ise 0 other/ise
x x x x
x x x x
= =
1
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= 1
10
10
= 2
= 3
10
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Distributions of functions of
Random Variablesamma distribution, 32distribution,
-onential distribution
Therorem
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LetX andY denote a indeendent random #ariables
each ha#ing a gamma distribution /ith arameters&,1' and &,2'. Then W(X 6 Y has a gamma
distribution /ith arameters &, 1 62'.
Proof:
( ) ( )
1 2
andX Y# t # t t t
= =
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1 2 1 2
t t t
+ = =
( ) ( ) ( )Therefore X Y X Y# t # t # t + =
Recogni8ing that this is the moment generating
function of the gamma distribution /ith arameters
&, 16 2' /e conclude that W(X 6 Y has a
gamma distribution /ith arameters &, 16 2'.
Therorem&e-tension to RV:s'
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Letx1,x2, " ,xdenote indeendent random #ariables each
ha#ing a gamma distribution /ith arameters &,"', " ( 1, 2, ",.
Then W(x16x26 " 6xhas a gamma distribution /ith
arameters &, 1 62 6" 6 '.
Proof:
( ) 1, 2...,
"
"x# t " t
= =
Therefore
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1 2 1 2 ...
...
t t t t
+ + + = =
( ) ( ) ( ) ( )1 2 1 2
... ...
x x x x x x# t # t # t # t + + + =
Recogni8ing that this is the moment generating
function of the gamma distribution /ith arameters
&, 16 2 6"6 n' /e conclude that
W(x16x2/ / xhas a gamma distribution /ith
arameters &, 16 2 6"6 '.
Therorem
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4uose thatxis a random #ariable ha#ing a
gamma distribution /ith arameters &,'.Then W( ax has a gamma distribution /ith
arameters &5a, '.
Proof:( )x# t
t
=
( ) ( )then ax x a# t # at at t
a
= = =
Special Cases
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1. LetX and Ybe indeendent random #ariables
ha#ing an e-onential distribution /ith arameter
then X 6 Y has a gamma distribution /ith ( 2
and 2. Letx1,x2,",x,be indeendent random #ariables
ha#ing a e-onential distribution /ith arameter then (x16x26"6xhas a gamma distribution
/ith ( and 3. Letx1,x2,",x,be indeendent random #ariables
ha#ing a e-onential distribution /ith arameter then
has a gamma distribution /ith ( and
1 x x0x
+ += = K
Distribution of
l i i l di ib i
x
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oulation = -onential distribution
Bnother illustration of the central limit theorem
L d b i d d d i bl
Special Cases !continued
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. LetX and Ybe indeendent random #ariables
ha#ing a2 distribution /ith 1 and 2 degrees of
freedom resecti#el% then X 6 Y has a2
distribution /ith degrees of freedom 1 6 2.
G. Letx1,x2,",x,be indeendent random #ariables
ha#ing a2 distribution /ith 1 ,2 ,", degreesof freedom resecti#el% thenx16x26"6xhas a
2 distribution /ith degrees of freedom 1 6"6 .
oth of these roerties follo/ from the fact that a2 random #ariable /ith degrees of freedom is a
random #ariable /ith ( and ( 52.
;f h 4 d d l di ib i h h
Recall
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;f z has a 4tandard ormal distribution thenz2 has a
2 distribution /ith 1 degree of freedom.
Thus ifz1,z2,",zare indeendent random #ariables
each ha#ing 4tandard ormal distribution then
has a2 distribution /ith degrees of freedom.
2 2 21 2 ...U z z z = + + +
Therorem
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4uose that U1and U2are indeendent random #ariables and
that U ( U16 U24uose that U1and Uha#e a2
distribution/ith degrees of freedom 1andresecti#el%. &1N '
Then U2has a2distribution /ith degrees of freedom 2(C1
Proof:
( )12
1
12
12
o/
(
U# tt
=
( )21
2
12
and
(
U# tt
=
( ) ( ) ( )Blso U U U# t # t # t=
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( ) ( ) ( )1 2
Blso U U U# t # t # t
2
12 2
12
12
1122
11 22
12
(
((
(
t
t
t
= =
( ) ( )( )2
1
7enceU
U
U
# t# t
# t=
Q.E.D.
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Tables for 4tandard ormal distri
bution
http://14%20s241%20use%20of%20normal%20tables.ppt/http://14%20s241%20use%20of%20normal%20tables.ppt/http://14%20s241%20use%20of%20normal%20tables.ppt/http://14%20s241%20use%20of%20normal%20tables.ppt/