, Moment To calculate the raw
Revision notes
Generating Function
g() variable Tallows us moments of a distribution from
Hy
= g(x)fx Standardising generate
=
a
the
-
calculatsents
( dos e
mgf We
Elg(1] to
ECY]
then
E(etY) the
=
variance = My(t) ECCEX
= .
o
transformation mean
=
calculating
about
Important when we are talking
mgt by
-
of variables X
* Seven a derivative
Note change end points forcummations
when Myt =
we then take the
and set
doing substitutions exponential to t
x)* e(t x)xdx with respect
. -
=
to make caculations
distributions
°
> Use (x t)x
-
x6
- -
-
=
0
da
.
= e
easier. Example changethe
look of the equation .
useful function gamma function
- Me
R
T(z)
= =
- -
-e d
-
= exp(tX + d
zu
Row Moments , Central Moments
Mate() x
exp)
26h
m) x +
-
-
2xm +
(oet "( a) -
E[XIT -
(c)
Raw moment -
E[X] -> first raw moment known as mean
(m)
a(X ++
0)"
'Cimmt
Central momet E[(X-M)"]
(0e
+
+ 1 -
(8 )
%
variance
ECCX-M72] + Second central moment known as
·
moments
Central moments can
be represented by raw
=
Y E2X& LECXS)"
ECX-MK]
-
=
M(x) (m! (x))
M(x)
-
=
2
Coefficient of skewness ↳
+
= E Mice
on exp(M) exp( ·
U, 30 position exp(mt
+
+ + 5) . exp(X
2 =
X ,
+ Xz + --
-
Xa t
1 + 6 %.
↳ exp (Mt
+
2+
U negea
mc
Me
-
-
Mz(t) =
2t)
-
-
2 m(l
-
-
=
2
et E M
el
-
-
=
of
CoficientKurI a
the limit
The poison distribution is
the binomial
>
-
Bin (n p
=
p(n))
,
e
lim up
-
pref Bin(n,n) = (n) poc-pi-
limpc-p-k =
Emm
How ?
variables
Mqfs and sums of random
Y
if 2 = X +
Mx(t) My(t) -
Mz(t) =
Mz(t) = Eletz]
+ (x
y)]
+
=
E[e
ECetX] ECCEYY
-
·
=
=
My()
-
My(t)
Distribution of the maximum and minimum
Maximum
P (max <X , .
.
. .
.
,
Xn][y)
P (for all i ,
Xi =
y)
=
P(X ,
EynXz(yn
- ..
nYnzy)
independence
=
PX < y .
. . .
.
-
PCXnEy)
Excel a
=ExyFx(y)--
.
Minimum
P(min (x ....
Xn) >
y)
= P (all j ,
Xi <
y)
= P(X )y1 , - . .
nXn]y)
Pay at
= PCX
1 ( Fx())"
Xn)(y)
-
-
=
P(min(x. ...
Revision notes
Generating Function
g() variable Tallows us moments of a distribution from
Hy
= g(x)fx Standardising generate
=
a
the
-
calculatsents
( dos e
mgf We
Elg(1] to
ECY]
then
E(etY) the
=
variance = My(t) ECCEX
= .
o
transformation mean
=
calculating
about
Important when we are talking
mgt by
-
of variables X
* Seven a derivative
Note change end points forcummations
when Myt =
we then take the
and set
doing substitutions exponential to t
x)* e(t x)xdx with respect
. -
=
to make caculations
distributions
°
> Use (x t)x
-
x6
- -
-
=
0
da
.
= e
easier. Example changethe
look of the equation .
useful function gamma function
- Me
R
T(z)
= =
- -
-e d
-
= exp(tX + d
zu
Row Moments , Central Moments
Mate() x
exp)
26h
m) x +
-
-
2xm +
(oet "( a) -
E[XIT -
(c)
Raw moment -
E[X] -> first raw moment known as mean
(m)
a(X ++
0)"
'Cimmt
Central momet E[(X-M)"]
(0e
+
+ 1 -
(8 )
%
variance
ECCX-M72] + Second central moment known as
·
moments
Central moments can
be represented by raw
=
Y E2X& LECXS)"
ECX-MK]
-
=
M(x) (m! (x))
M(x)
-
=
2
Coefficient of skewness ↳
+
= E Mice
on exp(M) exp( ·
U, 30 position exp(mt
+
+ + 5) . exp(X
2 =
X ,
+ Xz + --
-
Xa t
1 + 6 %.
↳ exp (Mt
+
2+
U negea
mc
Me
-
-
Mz(t) =
2t)
-
-
2 m(l
-
-
=
2
et E M
el
-
-
=
of
CoficientKurI a
the limit
The poison distribution is
the binomial
>
-
Bin (n p
=
p(n))
,
e
lim up
-
pref Bin(n,n) = (n) poc-pi-
limpc-p-k =
Emm
How ?
variables
Mqfs and sums of random
Y
if 2 = X +
Mx(t) My(t) -
Mz(t) =
Mz(t) = Eletz]
+ (x
y)]
+
=
E[e
ECetX] ECCEYY
-
·
=
=
My()
-
My(t)
Distribution of the maximum and minimum
Maximum
P (max <X , .
.
. .
.
,
Xn][y)
P (for all i ,
Xi =
y)
=
P(X ,
EynXz(yn
- ..
nYnzy)
independence
=
PX < y .
. . .
.
-
PCXnEy)
Excel a
=ExyFx(y)--
.
Minimum
P(min (x ....
Xn) >
y)
= P (all j ,
Xi <
y)
= P(X )y1 , - . .
nXn]y)
Pay at
= PCX
1 ( Fx())"
Xn)(y)
-
-
=
P(min(x. ...
