Solutions to Homework Set No. 6 – Probability Theory (235A), Fall 2011
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
dx
1
dx
2
. . . dx
n
→
n
→∞
f
(1
/
2)
.
Solution.
Let
X
1
, X
2
, . . .
be independent uniformly distributed random variables on [0
,
1].
E
f
S
n
n
=
Z
Ω
f
X
1
+
· · ·
+
X
n
n
d
P
=
Z
1
0
· · ·
Z
1
0
f
x
1
+
· · ·
+
x
n
n
dx
1
· · ·
dx
n
where we denote by Ω the product space and by
P
the probability measure on Ω. By the
SLLN,
S
n
n
→
1
2
a.s. and so does
f
(
S
n
n
→
f
(
1
2
)
a.s. because
f
is continuous. Now, by the
bounded convergence theorem we obtain the result. An alternative and more direct proof
method involves using the Weak Law of Large Numbers by following a method similar to
the proof of the Weierstrass approximation theorem using Bernstein polynomials.
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, Probability theory, Tn, Xn

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