Question

Problem 5 - Rare outcomes and data set size Here we will be concerned with a biased coin for which outcome 1 has a very low probability, i.e 0 < θι < 6o << 1. Assume our experiment consists of n independent tosses of this coin. 1. What is the probability po P(n1 0) that the outcome sequence contains no 1's? Write the answer as a function of θ| and n 2. What is the probability pi P(n1-1) that the outcome sequence contains a single 1, ? Write the answer as a function of θ| and n. 3. Note that for n 1 (and in general for small values of n) Po >> pi. We are interested in the value(s) of n for which po and pi are of comparable size. This gives us the answer to the question: how many times do I need to toss the coin in order to have a chance of observing the rare outcome once (pi) that is about the same to that of not observing it (Po)? To obtain the answer, solve the equation 0 for n a function of θ1. You will not necessarily obtain an integer 4. Compute the above n for 01 10-3, 10-4, 10-5

Answer 1

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(l-@j)n-ne,C1-Θ.)n-l (1-6) = ne, 仁白,+no, ラ ラ lo