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Independence (probability theory)#Two random variables :
X and Y with cumulative distribution functions
F
X
(
x
)
{\displaystyle F_{X}(x)}
F
Y
(
y
)
{\displaystyle F_{Y}(y)}
probability densities
f
X
(
x
)
{\displaystyle f_{X}(x)}
f
Y
(
y
)
{\displaystyle f_{Y}(y)}
iff the combined random variable (X , Y ) has a joint cumulative distribution function
F
X
,
Y
(
x
,
y
)
=
F
X
(
x
)
F
Y
(
y
)
,
{\displaystyle F_{X,Y}(x,y)=F_{X}(x)F_{Y}(y),}
or equivalently, if the joint density exists,
f
X
,
Y
(
x
,
y
)
=
f
X
(
x
)
f
Y
(
y
)
.
{\displaystyle f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y).}
How can one show that these are equivalent? Loraof (talk ) 03:41, 15 August 2017 (UTC)[ reply ]
F
X
,
Y
(
x
,
y
)
{\displaystyle F_{X,Y}(x,y)}
=
∫
−
∞
x
(
∫
−
∞
y
f
X
,
Y
(
x
,
y
)
d
y
)
d
x
{\displaystyle \int _{-\infty }^{x}\left(\int _{-\infty }^{y}f_{X,Y}(x,y)dy\right)dx}
=
∫
−
∞
x
(
∫
−
∞
y
f
X
(
x
)
f
Y
(
y
)
d
y
)
d
x
{\displaystyle \int _{-\infty }^{x}\left(\int _{-\infty }^{y}f_{X}(x)f_{Y}(y)dy\right)dx}
=
∫
−
∞
x
f
X
(
x
)
(
∫
−
∞
y
f
Y
(
y
)
d
y
)
d
x
{\displaystyle \int _{-\infty }^{x}f_{X}(x)\left(\int _{-\infty }^{y}f_{Y}(y)dy\right)dx}
=
(
∫
−
∞
x
f
X
(
x
)
d
x
)
(
∫
−
∞
y
f
Y
(
y
)
d
y
)
{\displaystyle \left(\int _{-\infty }^{x}f_{X}(x)dx\right)\left(\int _{-\infty }^{y}f_{Y}(y)dy\right)}
=
F
X
(
x
)
F
Y
(
y
)
{\displaystyle F_{X}(x)F_{Y}(y)}
Bo Jacoby (talk ) 11:19, 15 August 2017 (UTC).[ reply ] Loraof (talk ) 16:48, 15 August 2017 (UTC)[ reply ]
