independence
:
HW#9
E[X]
=
E[A]
+
E[B]
F(a
,
b)
=
F(a)
F(b)
-
check
both
sides
var(X]
=
Var[A]
+
Var(B]
+
2
COV(a
,
b)
f(x
,
y)
=
*
y
x
=
1
,
2
㉜
Y
:
1
,
2
.
3
.
h
E
[
XJ
:
XeF
(
x
)
y
1
.
0
.
2
+
2
.
0
.
3
If
theres
multiplication
find
marginals
82X
=
E[X
?
]
-
(E[X])
<
↳
5A
+
7
B
X
;
上世
な
+
け
3
+
世
=
管流
E(X
?
]
=
x
*.
F(X)
E
[
X
]
=
5
.
E
[
A
]
+
FEEEB
ECAB]
=
joint
probabilitye
from
tableline
nwo
Var(X]
=
↳
take
1
variable
S
'
VaVCAJ
+
T
'
VarBle
usncarcAin
)
“
“
×
(
1
)
=
(
1
.
y
)
.
p
(
l
.
y
)
-
>
summation
If
P(X(Y)
and
Its
integral
X
(
2
)
:
{
2
y
)
。
f
(
2
,
y
)
]
SSo"
dx
dy
or
↓
+initial
x
EXI
)
금다
)
+
용
3
)
=
2
.
y()
=
(X
.
1)
.
F(x
,
1)
]
summation
y
interval
E
]
豪
いる
←
⾼
(
に
)
+
(
3
)
+
いに
装
y
(2)
(X
.
2)
·
F(X
,
2)
To
Find
ECXY]
From
F(xy)
and
I
intervals
If
like
HW8
the
joint
probabis
used
to
↓
(X
.
y
.
f(x
,
y)dxdy
find
sume
AtB
=
X
use
the
new
probab
table
for
E(X]
02X
.
X
and
X2
poission
distribution
11
=
3
/2
=
4
.
5
Determine
the
value
of
C
to
make
F(xy)
↳
use
named
distributions
sheet
valid
joint
pruf
G
FAXY
に
C
(
X
+
Y
)
X
=
1
,
2
.
3
y
:
1
…
×
change
of
variables
X
1
に
1
り
=
こ
2
+
7
+
15
:
I
4
↳
Y
=
2X
e
geometric
X
12
)
:
(
+
1
)
2
+
(
2
+
2
)
=
7
C
=
リ
2
y
X
=
Y
2
qX
"
p
X
=
1
,
2
y
plug
into
X
13
)
:
13
+
17
+
(
3
+
2
)
+
13
+
3
)
=
15
DOUBLE
f(y)
=
qY/2
-1p
y
=
2
,
4
,
6
,
8
Y
に
2
X
*
If
X
and
y
are
intervals
take
integral
=
1)
change
of
Variabl es
-
absolute
value
marginal
pmf
values
from
above
over
fy
=
Fx(x)
.
(dX/dy)
or
if
integral
↳
check
if
mono
tone
に
X
2
F(x)
=
J
F(xy)
dy
on
y
interval
=
2x70
for
given
use
this
PDFe
CDF
gx
take
integral
·
f(y)
=
ff(x
y)dx
on
x
interval
also
for
example
:
CDF
+
PDF
F
(
X
)
=
4
X
3
Y
=
X
2
derivative
cov(a
,
b)
=
E(ab]
-
E(aJE
(b]
X
=
ry
corr(a
,
b)
=
corcaib)
Paf
:
fy
(
y
)
=
fx
(
ry
)
.
可
X
に
前
O
1
O
금
Fyly
)
=
4
(
ry
)
3
.
可
=
zy
ocyu
