Geometric distribution. PMF GEO(XIP) = (1-P-1 ; x = 1, 2, --- op1 (a) Calculate (derive)...

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Geometric distribution. PMF GEO(XIP) = (1-P²-1 ; x = 1, 2, --- o≤p≤1 (a) Calculate (derive) the Jeffreys prior for the geometric distribution parameter, p. Plot this distribution. Does it look "reasonable"? (b) By transformation of variables, find the prior distribution of p¹. Plot this distribution. Does it look "reasonable"? (c) Calculate (derive) the Jeffreys prior for p. That is, let u=p¹¹, set f(xlu) a u' (1-u), 1 ≤u<, and derive the Jeffreys prior for u. (d) How do the priors derived in (b) and (c) compare? Suppose a random sample of five observations, (11.4, 7.3, 9.8, 13.7, 10.6), are drawn from the Cauchy distribution (i.e., a t-distribution with 1 degree-of-freedom), p(y|0) = ¹ [1 + (y-0)²]-¹, -∞<y,0<∞0. (a) Assuming an uninformative (flat) prior on 0, derive the posterior distribution of 0 given the five observations. (b) Draw a graph of the posterior density. (c) Find the mean of the posterior density. (d) Calculate the P(0<11.5/data). (e) Find a 95% credible interval for the parameter 0. (b) Suppose the prior distribution for p is beta(1,1), and the "old" binomial experiment is based on 5 trials with 2 successes. The "new" experiment is to consist of 2 trials, so Xnew can take on values 0, 1, or 2. Write out the probability mass function for Xnew. Plot the pair of gamma probability densities and calculate the Kullback-Leibler distance (see Sec. 7.6 of Gill), based on the same data, in the following two cases (subscript 1 corresponds to f and subscript 2 corresponds to g): (a) a₁ =2, B₁ = 5; 02 = 2, B₂ = 15 (b) a₁ = 2, B₁= 5; 02= 10, B₂=5. Geometric distribution. PMF GEO(XIP) = (1-P²-1 ; x = 1, 2, --- o≤p≤1 (a) Calculate (derive) the Jeffreys prior for the geometric distribution parameter, p. Plot this distribution. Does it look "reasonable"? (b) By transformation of variables, find the prior distribution of p¹. Plot this distribution. Does it look "reasonable"? (c) Calculate (derive) the Jeffreys prior for p. That is, let u=p¹¹, set f(xlu) a u' (1-u), 1 ≤u<, and derive the Jeffreys prior for u. (d) How do the priors derived in (b) and (c) compare? Suppose a random sample of five observations, (11.4, 7.3, 9.8, 13.7, 10.6), are drawn from the Cauchy distribution (i.e., a t-distribution with 1 degree-of-freedom), p(y|0) = ¹ [1 + (y-0)²]-¹, -∞<y,0<∞0. (a) Assuming an uninformative (flat) prior on 0, derive the posterior distribution of 0 given the five observations. (b) Draw a graph of the posterior density. (c) Find the mean of the posterior density. (d) Calculate the P(0<11.5/data). (e) Find a 95% credible interval for the parameter 0. (b) Suppose the prior distribution for p is beta(1,1), and the "old" binomial experiment is based on 5 trials with 2 successes. The "new" experiment is to consist of 2 trials, so Xnew can take on values 0, 1, or 2. Write out the probability mass function for Xnew. Plot the pair of gamma probability densities and calculate the Kullback-Leibler distance (see Sec. 7.6 of Gill), based on the same data, in the following two cases (subscript 1 corresponds to f and subscript 2 corresponds to g): (a) a₁ =2, B₁ = 5; 02 = 2, B₂ = 15 (b) a₁ = 2, B₁= 5; 02= 10, B₂=5.

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