You have 2 coins and a spinning pointer U. The coins are fair and unbiased, and the pointer U has a uniform distribution over [0, 1). You flip the both coins and spin the pointer. A random variable X is defined as follows: If the first coin is "heads", then:
If the first coin is "tails", then X = U + 2. Define another random variable:
(a) Find FX(x).
(b) Find P r( ½ ≤ X ≤ 2½ ).
(c) Sketch the pdf of Y and label important values.
(d) Design an optimal detection rule to estimate U if you are given only Y. What is the probability of error?
(e) State how to, or explain why it is not possible to:
i. Generate a binary random variable Z, pZ(1) = p, given U?
ii. Generate a continuous, uniformly distributed random variable given Z?