The basic idea behind independence is that if your random variables (or vectors) are independent then the combination of several of these random variable/vectors
(
P
(
X
1
≤
x
1
,
.
.
.
,
X
d
≤
x
d
)
)
{\displaystyle \left(P\left(X_{1}\leq x_{1},...,X_{d}\leq x_{d}\right)\right)}
Given
T
1
,
T
2
{\displaystyle T_{1},T_{2}\,}
λ
-Exponential
{\displaystyle \lambda \,{\mbox{-Exponential}}}
E
[
T
1
+
T
2
]
=
E
[
2
T
1
]
=
E
[
2
T
2
]
{\displaystyle E\left[T_{1}+T_{2}\right]=E\left[2T_{1}\right]=E\left[2T_{2}\right]}
Var
(
T
1
+
T
2
)
<
Var
(
2
T
1
)
{\displaystyle {\mbox{Var}}\left(T_{1}+T_{2}\right)<{\mbox{Var}}\left(2T_{1}\right)}
Var
(
T
1
+
T
2
)
=
Var
(
T
1
)
+
Var
(
T
2
)
=
2
Var
(
T
1
)
{\displaystyle {\mbox{Var}}\left(T_{1}+T_{2}\right)={\mbox{Var}}\left(T_{1}\right)+{\mbox{Var}}\left(T_{2}\right)=2{\mbox{Var}}\left(T_{1}\right)}
Var
(
2
T
1
)
=
2
2
⋅
Var
(
T
1
)
{\displaystyle {\mbox{Var}}\left(2T_{1}\right)=2^{2}\cdot {\mbox{Var}}\left(T_{1}\right)}
See the equations section for some more examples.
P
(
X
1
≤
x
1
,
.
.
.
,
X
d
≤
x
d
)
{\displaystyle P\left(X_{1}\leq x_{1},...,X_{d}\leq x_{d}\right)\,}
=
P
(
X
1
≤
x
1
)
⋯
P
(
X
d
≤
x
d
)
{\displaystyle =P\left(X_{1}\leq x_{1}\right)\cdots P\left(X_{d}\leq x_{d}\right)\,}
F
X
1
,
X
2
,
.
.
.
X
d
(
x
1...
d
)
{\displaystyle F_{X_{1},X_{2},...X_{d}}(x_{1...d})\,}
=
F
X
1
(
x
1
)
⋯
F
X
d
(
x
d
)
{\displaystyle =F_{X_{1}}(x_{1})\cdots F_{X_{d}}(x_{d})\,}
f
X
1
,
X
2
,
.
.
.
X
d
(
x
1...
d
)
{\displaystyle f_{X_{1},X_{2},...X_{d}}(x_{1...d})\,}
=
f
X
1
(
x
1
)
⋯
f
X
d
(
x
d
)
{\displaystyle =f_{X_{1}}(x_{1})\cdots f_{X_{d}}(x_{d})\,}
E
[
X
1
⋅
X
2
⋯
X
d
]
{\displaystyle E\left[X_{1}\cdot X_{2}\cdots X_{d}\right]\,}
=
E
[
X
1
]
⋯
E
[
X
d
]
{\displaystyle =E\left[X_{1}\right]\cdots E\left[X_{d}\right]\,}
E
[
g
1
(
X
1
)
⋅
g
2
(
X
2
)
⋯
g
d
(
X
d
)
]
{\displaystyle E\left[g_{1}\left(X_{1}\right)\cdot g_{2}\left(X_{2}\right)\cdots g_{d}\left(X_{d}\right)\right]\,}
=
E
[
g
1
(
X
1
)
]
⋯
E
[
g
d
(
X
d
)
]
{\displaystyle =E\left[g_{1}\left(X_{1}\right)\right]\cdots E\left[g_{d}\left(X_{d}\right)\right]\,}
M
X
1
+
.
.
.
+
X
d
(
x
)
{\displaystyle M_{X_{1}+...+X_{d}}(x)\,}
=
M
X
1
(
x
)
⋯
M
X
1
(
x
)
{\displaystyle =M_{X_{1}}(x)\cdots M_{X_{1}}(x)\,}
F
X
Y
(
x
,
y
)
{\displaystyle F_{XY}(x,y)\,}
=
?
?
?
{\displaystyle =???\,}
f
X
Y
(
x
,
y
)
{\displaystyle f_{XY}(x,y)\,}
=
?
?
?
{\displaystyle =???\,}
F
X
|
Y
(
x
|
y
)
{\displaystyle F_{X|Y}(x|y)\,}
=
?
?
?
{\displaystyle =???\,}
F
X
+
Y
(
x
)
{\displaystyle F_{X+Y}(x)\,}
=
P
(
X
+
Y
≤
x
)
{\displaystyle =P\left(X+Y\leq x\right)\,}
![{\displaystyle E\left[T_{1}+T_{2}\right]=E\left[2T_{1}\right]=E\left[2T_{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/599751fbffc23be31289b44eba3a1ad9a0133280)
![{\displaystyle E\left[X_{1}\cdot X_{2}\cdots X_{d}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bef5a4c4237aad4feec73fd7efc01b56b11cfc1)
![{\displaystyle =E\left[X_{1}\right]\cdots E\left[X_{d}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1b907abc4c18e35c2c6483358a89f466adfa09)
![{\displaystyle E\left[g_{1}\left(X_{1}\right)\cdot g_{2}\left(X_{2}\right)\cdots g_{d}\left(X_{d}\right)\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37550f7b354be2899133a438e3d3c6d6193007b7)
![{\displaystyle =E\left[g_{1}\left(X_{1}\right)\right]\cdots E\left[g_{d}\left(X_{d}\right)\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29dd5d649f7cf8d1f0bf8ae83dfa30fe8718b194)
