Transformations
Find
Fyly) Fy(y) fx(x) **g) :
=
Ex) , /
·
9
&
·
weight
th
Ex) /x) "Hunchanged
-
Y=
1) Calc. ratio by
diff both
Distribution
of weight
+ HisCstasa
sides Y4
2(+) =
~
where it overlaps A bh= =
y Ex 1
+ =
-
dy =d yex = keeps shape - - 3
weight &
-
2 2 -
2) for
8
5 3
sub fx(x) jπ(*)(fy(y) Em)"j) + S(y 2)
Solve &
-
x
[S(y) +S(y 1) +π(y 1) &
fyly)
=
=
-
-
= + -
,
[x +
·· ·
y= 1 + X = 2y 2
Ex)
-
12
Y x = Ex)3/2
n
=
π() 5S(x 1)
=
fx(x) = = + +
ty
2 3/2
-
-
becomes b
o
fy(y) = [5π)(2) 55S((2y +
2) + 1)) (2)
·
14
weight it "
-
= π()
"I
m
+
5S(y -)
weight 2 (ii) ·
-
2 2
(weight untouched a
=
chart intation stunct approx
fy(y) jjπ(() fy(y) =S(y) [S(y z)
use
[S(y 4)
. ·
[(x) + 1 = iS(y)
+ + +
- -
=
X = +1 + y =
fx(x) =(e)
Calculations EX1) nY2
Ex2) Ys
I
Expectation Operator Statistical X-center l
width
E24] =
22g(x)] -Syfy(yidy / g(x)fx(x)dx =
-
I I
nangero
Mean-Simplest Transformation
should be Of 11X11
My weight Location
+ :
=
mean
-measures balance pt ·
of PDF
distribution
(π(*) [S(x 2)]ax
/[ π()]dx
+
-center
of gravity of
-
=
m
mean= 2(x) =
:Xf(x)dx E(X]
= x()ax I
Variance = Measures
mament fine = (E) ! .
= (j) + (2) var(x) =
5/3
small clustered around
Large- spanned mean out =+ (1 1) -
= or
= I
E(X-mil =(x-mifx(x)dx 0 20 * (i)]ax =x[() -
2[x] = E[x] ES(x 2)]dx
=
+ -
alternate form to = /, xdx
2(x-mY 2[x] mi E(x*] (2[x])
"
[] ! = x()dx +(2
·
=
-
=
-
=
Std
.Deviationhassameunitsaits
6" = E[x] -
m2 = O -
i
Meg
sta dev 6 =
=
&
(x)
=
6 .
-
fx(x) Ex3) pl
kabe
where
many
mode -
is max Area under curve
-value
of X at peath(s) for PDFmust 1 =
edi
g play role in
salvefox fx(x) St
=
-
RV X
o
nonnegative
-
=
fi
mean calc
g apgs(x 12
.
Multimodal but & median mean =
g(0) 2pg(1) p2(2)
+ +
only I J
mean
ps(x 2)
·
+ +
2p([ + p) 1 for binomial dist.
- -
= =
Median-center weight point A median where
2p
=
, ,
formarbitraryare
2 :
n
g(0) + 2pg(1)) + p2(2)
-max
E[x] m2
=
=
var = -
=
psusFunivax
K 0
, 1 2
2pq + 4p3 (2p)2
=
= -
.
=
2pq
P(X = X B)
Fx(x(B) P(x = x(B) CDF RHS 1
a
= = , -
=
P(13)
1) *
S in
B =
LasX : b] const a b G X,
i
.
,
Il
P(X = X acX= b) Fx(x) Fx(a)
P(X x(a
-
F(x(axX = b) X = b) &
,
=
= = + =
p(a(X = b) Fx(b) Fx(a) -
2) B =
[X = b] const .
b drive a => -@
I X>b
G b(fx(x)dx
x =
P(X =, X =)
F(x(X = b) =
P(x = x(x = b) =
=
P(X = b) X7 b
I
·
3)
B= [Xa] cast. a driveb >
-
G
X1a
e h
changes
E(x(X(a) =
P(X x(X) =
=
Pl) X>a
&
-
I
Ex1
find
chebyshev
upper bound
p((x-m1 6) &
104
goo loo no e >
T i:
m 1000 4 + =
s ⑪
=
-X -
m = kG
6 50 1 (50) 100 P((x mi > 26) 90
IX
= = -
an
e
·
m = 1 ↳
r
-
2 k 2
k=
P((x u(zx))
=
T D
-
↑ ((x 100012 100)
- <
i = 0 .
25 =
25 %
P((x-100012501) ! The
at most 25 % resistersWe discarded
)
=
Find
Fyly) Fy(y) fx(x) **g) :
=
Ex) , /
·
9
&
·
weight
th
Ex) /x) "Hunchanged
-
Y=
1) Calc. ratio by
diff both
Distribution
of weight
+ HisCstasa
sides Y4
2(+) =
~
where it overlaps A bh= =
y Ex 1
+ =
-
dy =d yex = keeps shape - - 3
weight &
-
2 2 -
2) for
8
5 3
sub fx(x) jπ(*)(fy(y) Em)"j) + S(y 2)
Solve &
-
x
[S(y) +S(y 1) +π(y 1) &
fyly)
=
=
-
-
= + -
,
[x +
·· ·
y= 1 + X = 2y 2
Ex)
-
12
Y x = Ex)3/2
n
=
π() 5S(x 1)
=
fx(x) = = + +
ty
2 3/2
-
-
becomes b
o
fy(y) = [5π)(2) 55S((2y +
2) + 1)) (2)
·
14
weight it "
-
= π()
"I
m
+
5S(y -)
weight 2 (ii) ·
-
2 2
(weight untouched a
=
chart intation stunct approx
fy(y) jjπ(() fy(y) =S(y) [S(y z)
use
[S(y 4)
. ·
[(x) + 1 = iS(y)
+ + +
- -
=
X = +1 + y =
fx(x) =(e)
Calculations EX1) nY2
Ex2) Ys
I
Expectation Operator Statistical X-center l
width
E24] =
22g(x)] -Syfy(yidy / g(x)fx(x)dx =
-
I I
nangero
Mean-Simplest Transformation
should be Of 11X11
My weight Location
+ :
=
mean
-measures balance pt ·
of PDF
distribution
(π(*) [S(x 2)]ax
/[ π()]dx
+
-center
of gravity of
-
=
m
mean= 2(x) =
:Xf(x)dx E(X]
= x()ax I
Variance = Measures
mament fine = (E) ! .
= (j) + (2) var(x) =
5/3
small clustered around
Large- spanned mean out =+ (1 1) -
= or
= I
E(X-mil =(x-mifx(x)dx 0 20 * (i)]ax =x[() -
2[x] = E[x] ES(x 2)]dx
=
+ -
alternate form to = /, xdx
2(x-mY 2[x] mi E(x*] (2[x])
"
[] ! = x()dx +(2
·
=
-
=
-
=
Std
.Deviationhassameunitsaits
6" = E[x] -
m2 = O -
i
Meg
sta dev 6 =
=
&
(x)
=
6 .
-
fx(x) Ex3) pl
kabe
where
many
mode -
is max Area under curve
-value
of X at peath(s) for PDFmust 1 =
edi
g play role in
salvefox fx(x) St
=
-
RV X
o
nonnegative
-
=
fi
mean calc
g apgs(x 12
.
Multimodal but & median mean =
g(0) 2pg(1) p2(2)
+ +
only I J
mean
ps(x 2)
·
+ +
2p([ + p) 1 for binomial dist.
- -
= =
Median-center weight point A median where
2p
=
, ,
formarbitraryare
2 :
n
g(0) + 2pg(1)) + p2(2)
-max
E[x] m2
=
=
var = -
=
psusFunivax
K 0
, 1 2
2pq + 4p3 (2p)2
=
= -
.
=
2pq
P(X = X B)
Fx(x(B) P(x = x(B) CDF RHS 1
a
= = , -
=
P(13)
1) *
S in
B =
LasX : b] const a b G X,
i
.
,
Il
P(X = X acX= b) Fx(x) Fx(a)
P(X x(a
-
F(x(axX = b) X = b) &
,
=
= = + =
p(a(X = b) Fx(b) Fx(a) -
2) B =
[X = b] const .
b drive a => -@
I X>b
G b(fx(x)dx
x =
P(X =, X =)
F(x(X = b) =
P(x = x(x = b) =
=
P(X = b) X7 b
I
·
3)
B= [Xa] cast. a driveb >
-
G
X1a
e h
changes
E(x(X(a) =
P(X x(X) =
=
Pl) X>a
&
-
I
Ex1
find
chebyshev
upper bound
p((x-m1 6) &
104
goo loo no e >
T i:
m 1000 4 + =
s ⑪
=
-X -
m = kG
6 50 1 (50) 100 P((x mi > 26) 90
IX
= = -
an
e
·
m = 1 ↳
r
-
2 k 2
k=
P((x u(zx))
=
T D
-
↑ ((x 100012 100)
- <
i = 0 .
25 =
25 %
P((x-100012501) ! The
at most 25 % resistersWe discarded
)
=
