Suppose that we want to generate a random variable X that is equally likely to be either 0 or 1, and that all we have at our disposal is a biased coin that, when flipped, lands on heads with some (unknown) probability p. Consider the following procedure: 1. Flip the coin, and let 21, either heads or tails, be the result.
2. Flip the coin again, and let Z2 be the result.
3. If Z and Z, are the same, return to step 1.
4. If Z, is heads, set X = 0, otherwise set X = 1.
a) Let I, and I be independent indicator variables with success probability p. Compute P(11 = 1|11 # 12)
b) Show that the random variable X generated by this procedure is equally likely to be either 0 or 1.
c) Could we use a simpler procedure that continues to flip the coin until the last two flips are different, and then sets X = 0 if the final flip is a head, and sets X = 1 if it is a tail?
PROBLEM 4.7
Implement the algorithm described in the previous problem in R and use it to generate 10 standard uniformly distributed variables.