The pdf, the cf, and the mgf of a random variable
• The following FT and LT relation among pdf, cf, and mgf can be utilized to
find the pdf of “the sum of many independent random variables”
• For discrete rvs, the “integration” is replaced by the “summation”
For a given a rv
,
its robability density function (pdf),
( )
the characteristic function (cf),
(f),
and the moment generation function (mgf),
(s)
are connected by FT and LT as follows:
(f)
X
X
X
X
X
fx
Φ
Φ
Φ
±
{} {}
22
(
)
()
(
s
)
jf
X
x
XX
sX
sx
X
E
ee
f
x
d
x
F
T
f
x
Ee
e
f
xd
x
L
T
f
x
ππ
+∞
⋅⋅
+
−∞
+
=⋅
=
Φ=
⋅
=
∫
∫
±
{ }
{}
2
2
(f)
(
)
(s)
(
)
n
n
x
X
Xn
n
n
e
PX x
e PX x
π
⋅
⋅
⋅
⋅
=
=
∑
∑
±
±
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FACT: the pdf of “the sum of independent random variables” is the
convolution
of all the pdfs of the independent rvs contained in the
summation.
12
N
n=1
If
=
, with
( ) and
' are independent,
t
h
e
p
d
f
o
f
()
,
a
n
d
the cf of
:
( )
( )
( )
( ),and
the mgf of
:
( )
( )
( )
( )
If we furthe
n
n
n
n
nn
X
n
YXX
X
YX
X
X
X
X
X
f
x
X
s
Yf
y
fyfy
fy
f
f
f
Ys
s
s
s
∼
∼=
∗
∗
∗
Φ=
Φ
⋅
Φ
⋅
⋅
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 Spring '11
 HONGYAGE
 Central Limit Theorem, Normal Distribution, Variance, Probability distribution, Probability theory

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