Bill tosses a bent coin N times, obtaining a sequence of heads and tails. We ass
ID: 3207017 • Letter: B
Question
Bill tosses a bent coin N times, obtaining a sequence of heads and tails. We assume that the coin has a probability f_H of coming up heads; we do not know h_H. If n_H heads have occurred in N tosses, what is the probability distribution of f_H? (For example, N might be 10, and n_H might be 3; or, after a lot more tossing, we might have N = 300 and n_H = 29.) What is the probability that the N + 1th outcome will be a head, given n_H heads in N tosses? Assuming a uniform prior on f_H, P(f_H) = 1, solve the problem posed in example 2.7 (p.30). Sketch the posterior distribution of f_H and compute the probability that the N + 1th outcome will be a head, for N = 3 and n_H = 0; N = 3 and n_H = 2; N = 10 and n_H = 3; N = 300 and n_H = 29. You will find the beta integral useful: integral_0^1 d p_a p_a^Fa (1 - p_a)^F_b = Gamma(F_a + 1)Gamma(F_b + 1)/Gamma(F_a + F_b + 2) = F_a!F_b!/(F_a + F_b + 1)! (2.32)Explanation / Answer
2.7)
coin tossed=N times
no of heads= nH
probabilty = fH
the probabilty distribution will be binomial distribution
P(nH/fH,N)= NCnH*fH^nH*(1-fH)^(N-nH)
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