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One can conclude from this that
f
X,Y
(
x,y
) =
f
X
(
x
)
f
Y
(
y
)
,
or again the joint density factors. Going the other way, one can also see that if the joint density factors,
then one has independence.
Example.
Suppose one has a ﬂoor made out of wood planks and one drops a needle onto it. What is the
probability the needle crosses one of the cracks? Suppose the needle is of length
L
and the wood planks are
D
across.
Answer.
Let
X
be the distance from the midpoint of the needle to the nearest crack and let Θ be the angle
the needle makes with the vertical. Then
X
and Θ will be independent.
X
is uniform on [0
,D/
2] and Θ is
uniform on [0
,π/
2]. A little geometry shows that the needle will cross a crack if
L/
2
> X/
cos Θ. We have
f
X,
Θ
=
4
πD
and so we have to integrate this constant over the set where
X < L
cos Θ
/
2 and 0
≤
Θ
≤
π/
2
and 0
≤
X
≤
D/
2. The integral is
Z
π/
2
0
Z
L
cos
θ/
2
0
4
πD
dxdθ
=
2
L
πD
.
If
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This note was uploaded on 12/29/2011 for the course MATH 317 taught by Professor Wen during the Spring '09 term at SUNY Stony Brook.
 Spring '09
 wen
 Factors

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