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Mathematical Statistics
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Unit1Random variables,
Distribution functions,and Expectation
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Definition: A random variable (r.v.) is a function, whichassigns to each sample element a real number.
Example: Consider the experiment of tossing a single coin.
So the sample space S={head, tail}. Let the
random variable X denote the number of heads.
So define X(c)=1 if c=head, and X(c)=0 if c =
tail.
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Definition: Discrete random variable: A random
variable X will be defined to be discrete if
the range ofXis countable.
Example:
Example:
{ }
)(6
1)( 6,,3,2,1 xIxfX L=
{ } )(2
1)( 1,0 xIxfX =
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Definition: If X is a discrete r.v. with distinct values
then the probability density(mass) function (p.d.f. or p.m.f.) is defined
by
Example: If then
,,,,, 21LL
nxxx
LL ,,,,),()( 21 nX xxxxxXPrxf ="==
{ }
),(6
1)( 6,,2,1 xIxfX L= =)1(Xf
6
1)1( ==XPr
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Theorem: A function is a p.d.f. of a r.v. X if and
only if (iff)
a.
b.
Example:1.Findksuch that
is ap.d.f.
2. Findksuch that
is ap.d.f.
)(xfX
,1)(0 xfX x"
=xX xf 1)(
{ }
)()2
1
()( ,2,1 xIkxfx
X L=
{ } 10),()( ,1,0
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Definition: The (cumulative) distribution function
(c.d.f.) of a random variable X, denoted by
is defined to be for
every real numberx.
),(XF )()( xXPrxFX =
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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RemarkThe properties of :
1.
2. for
(i.e. is a nondecreasing function )
3.
4.
5. where is the left-
hand limit of at
6. is continuous from the right.
)(XF
,0)( =-XF 1)( =XF
)()( bFaF XX .ba
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Example: Let
(a) Find
(b) Find
(c) Find
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Example: Let
(a) Find
(b) Find
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Definition: Continuous random variableA r.v. X iscalled continuous if there exists a function
such that for
every real numberx.
Definition: The p.d.f. of a continuous r.v. X is the
function that satisfies
)(Xf -=x
XX duufxF )()(
.,)()( xduufxFx
X "= -
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Theorem: A function is a p.d.f. of a continuous
r.v. Xiff
a.
b.
Example: Findksuch that is a
p.d.f.
Example: Findksuch that is a
p.d.f.
)(xfX
xxfX " ,0)(
- =1)( dxxfX
)()( ),( xIkexfux
X ---
=
)()( ),0(2 xIekxxf xX
- =
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Remark: IfXis a continuousr.v., then
(a)
(b)
(c)
)()( xfxFdxd XX =
=
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Example: Let Define
Find thep.d.f.ofY.
Example: LetX~ Define
Find thep.d.f.ofY.
Example: Let Define Find
thep.d.f.ofY.
).()(~ )1,0( xIxfX X = .ln2 XY -=
).(100
1)( )100,0( xIxfX = XY 10=
).()(~ )1,0( xIxfX X = .XY =
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Definition: The expected value or mean of ar.v.
denoted by is
ifXis continuous
ifXis discrete
),(Xg
[ ],)(XgE
[ ]
,)()()(
-= dxxfxgXgE X
,)()(=x
X xfxg
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Example: LetX Find
Example: LetX~ Find
Example: LetX~ Find
Example: LetX~ Find
Example: Let Find
).(~ lpoisson ).(XE
).,( pnB ).(XE
).,(2
smN ).(XE
{ } ).(2
1)( ...,2,1 xIxf
x
X
= ).(XE
).(1
11)(~ ),(2 xIxxfX X -
+=p
).(XE
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Remark: Let X be a r.v. and let a, b be constants. Then
for any functions and whoseexpectations exist,
)(1 Xg )(2 Xg
bXgEaxbXga
XgExXg
XgbEXgaEXbgXagE
XgaEXagE
bbE
+=+
=
=
)]([then,allfor)(If.5
0)]([then,allfor0)(If.4
)]([)]([)]()([.3
)]([)]([.2
)(.1
11
11
2121
11
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Definition: A mode of a distribution of one r.v. X of
the continuous or discrete type is a value of
xthat maximizes thep.d.f.
Example: Find the mode of each of the following
distributions
).(xfX
{ }
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Definition: A (100p)th percentile of the distribution of a
r.v. X is a valueksuch thatand
Remark:1. A median of a r.v. X is a 50th percentile.
2. If X is a continuous r.v., then
(a) the (100p)th percentile of X is k satisfying
and
(b) the median ofXismsatisfying
pkXPr
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Example: Find the median of the following distributions:
zero elsewhere
ExampleFind the 20th percentile of
)()1(3)()( )1,0(2 xIxxfa X -=
,42or10,3
1)()(
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Homework:
1. Find the median of the distribution
where
2. Find the mode of the distribution
3. If then find the median.
4. Let X be a continuous nonnegative r.v. and
where and are
constants, Find
),(1
)( ),(
)(
xIexf
x
X qlq
l
q
--
= .,0 l,a m
.0,0,10 >>
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Definition: For each integer n the nth moment of X, is
defined as and the nth centralmoment ofX, is defined as
Definition: The variance of a r.v. X is its second centralmoment, The positive
square root of is the standard
deviation, ofX.
)(n
XE.)(
nEXXE -
.)()( 2EXXEXVar -=2)( s=XVar
,s
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Remark:1.
2. where a is a constant.
3. where a , b are
constants.
22 )()( EXEXXVar -=
,0)( =aVar
),()( 2 XVarabaXVar =
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Example: Find the variance of the following distributions:
{ } )(2
1)()(
)()(
),()(
),()(
,2,1 xIxfd
poissonc
pnBb
Gammaa
x
X L
=
l
ba
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Definition: Let X be a r.v. with c.d.f. The
moment-generating function (m.g.f.) of X,denoted by is
provided that expectation exists for
Theorem: IfXhasm.g.f. then
).(XF
),(tMX ),()(tX
X eEtM =
.0, >
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Example: Find for each of the following p.d.f.:)(tMX
{ } )(2
1)()(
)()(
),()(
)()()(
,2,1
),0(
xIxfd
poissonc
pnBb
xIexfa
x
X
xX
L
=
= -
l
l l
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Example: Let and define Find
thep.d.f.ofY, and
Example: Let and define Findthe p.d.f. of Y. (Please use c.d.f. method and
m.g.f.method)
)1,0(~UX .ln2 XY -=
)(YE ).(YVar
),(~
2
smNX .s
m-=
X
Y
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Example: LetXbe ar.v.such that
and
Find thep.d.f.ofX.
Example: Find the moments of the distribution that
hasm.g.f.
,!2
)!2()( 2
m
mXE
m
m
=
,0)( 12 =-mXEK,3,2,1=m
.1,)1()( 3
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Definition: If X is a r.v., the rth factorial moment of X
is defined as
Definition: The factorial moment generating function
of a r.v. X is defined (if it exists) as
)0( >r
[ ].)1()1( +-- rXXXE K
).()( XX tEtm =
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Remark: where we define
Example: Let Find
Example: Let Find
[ ]
),1()1()1( )(rXmrXXXE =+-- K
.|)()1( 1)( == tXr
r
rX tm
dtdm
).(~ lpoissonX [ ].)2)(1( -- XXXE
).,(~ pnBX ).(tmX
Text book: 1. Introduction to the Theory of Statistics by Mood Graybill et. al.2. Introduction to Mathematical Statistics by Robert V. Hogg & Allen T. Craig
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Theorem: Let X be a r.v. and a nonnegative
function with domain the real line then
for every
Corollary: Chebyshev inequality. If X is a r.v. with
finite variance, then
for every or
)(g
[ ][ ]
,)(
)(k
XgEkXgPr .0>k
[ ]2
1
rrXPr - sm 0>r
[ ] .112r
rXPr -
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Example: IfXis ar.v.such that and
use the Chebyshev inequality to determine alower bound for
Example: Does there exist a r.v. X for which
3)( =XE ,13)( 2 =XE
[ ].82
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Homework:
1. Let Prove
2.LetXbe ar.v.withm.g.f. Prove that
and
).,(~ baGammaX .2
)2(
a
ab
e
XPr
.),( hthtMX