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Homework Set No. 6 – Probability Theory (235A), Fall 2009
Posted: 11/3/09 — Due: 11/10/09
1.
Let
f
: [0
,
1]
→
R
be a continuous function. Prove that
Z
1
0
Z
1
0
...
Z
1
0
f
±
x
1
+
x
2
+
...
+
x
n
n
²
dx
1
dx
2
...dx
n
→
n
→∞
f
(1
/
2)
.
2.
A bowl contains
n
spaghetti noodles arranged in a chaotic fashion. Bob performs the
following experiment: he picks two random ends of noodles from the bowl (chosen uniformly
from the 2
n
possible ends), ties them together, and places them back in the bowl. Then he
picks at random two more ends (from the remaining 2
n

2), ties them together and puts
them back, and so on until no more loose ends are left.
Let
L
n
denote the number of
spaghetti loops
at the end of this process (a loop is a
chain of one or more spaghettis whose ends are tied to each other to form a cycle). Compute
E
(
L
n
) and
V
(
L
n
). Find a sequence of numbers (
b
n
)
∞
n
=1
such that
L
n
b
n
P
→
n
→∞
1
,
if such a sequence exists.
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 Fall '11
 DanRomik
 Probability

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