If 7.3. X is uniform-[0, 1], then E e If Y is uniform=[b, a + b], then E e
itY it(aX+b)
=
1 0
e
it(ax+b)
eit(a+b) - eitb dx = . iat
=
a+b eity b
eit(a+b) - eitb dy = . a iat
The uniqueness theorem does the rest. Next suppose Z is uniform-[0, 1]. Then Z = 2-i Zi where the Zi 's i=1 are 1 i.i.d. with values in {0, 1} with probability 2 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 ,
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