Download ∑ p. 38 7. The Central Limit Theorem e

7.3. If X is uniform-[0, 1], then
If Y is uniform=[b, a + b], then
!
" #1
eit(a+b) − eitb
it(aX+b)
it(ax+b)
E e
e
dx =
.
=
iat
0
$ itY % #
E e
=
b
a+b eity
eit(a+b) − eitb
dy =
.
a
iat
−i
The uniqueness theorem does the rest. Next suppose Z is uniform-[0, 1]. Then Z = ∑∞
i=1 2 Zi where the Zi ’s are
i.i.d. with values in {0, 1} with probability 21 each. From Problem 1.15 we know that X = 2Z − 1 is uniform[−1, 1]. But
∞
∞
∞
X = ∑ 2 2Zi − 1 = ∑ 2 (2Zi − 1) = ∑ 2−i Xi ,
i=1
−i
i=1
−i
i=1
and it follows easily that the Xi ’s are i.i.d. taking values ±1 with probability
1
2
each.