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{"ops":[{"insert":"Suppose an engineer is 95% confident that the probability of rejecting a product is going to be 0"},{"attributes":{"italic":true},"insert":"."},{"insert":"5 \u00b1 0"},{"attributes":{"italic":true},"insert":"."},{"insert":"2.\nUse this information to construct a beta prior for\u00a0"},{"attributes":{"italic":true},"insert":"\u03b8"},{"insert":"."},{"attributes":{"list":"ordered"},"insert":"\n"},{"insert":"Hint.\u00a0Use normal approximation for the beta distribution. Note that if\u00a0"},{"attributes":{"italic":true},"insert":"X"},{"insert":"\u00a0\u223c B"},{"attributes":{"italic":true},"insert":"eta"},{"insert":"("},{"attributes":{"italic":true},"insert":"\u03b1,\u03b2"},{"insert":"), then and.\nSuppose that the engineer observed 4 rejections among 10 products. Find the posterior distribution for\u00a0"},{"attributes":{"italic":true},"insert":"\u03b8"},{"insert":"\u00a0using the prior you proposed in (a)."},{"attributes":{"list":"ordered"},"insert":"\n"},{"insert":"Problem 2.\u00a0Suppose the posterior distribution of\u00a0"},{"attributes":{"italic":true},"insert":"\u03b8"},{"insert":"\u00a0is given by a mixture normal distribution:\u00a0"},{"attributes":{"italic":true},"insert":"\u03b8"},{"insert":"|"},{"attributes":{"italic":true},"insert":"data"},{"insert":"\u00a0\u223c 0"},{"attributes":{"italic":true},"insert":"."},{"insert":"5 \u00b7 N(0"},{"attributes":{"italic":true},"insert":","},{"insert":"1) + 0"},{"attributes":{"italic":true},"insert":"."},{"insert":"5 \u00b7 N(10"},{"attributes":{"italic":true},"insert":","},{"insert":"1)"},{"attributes":{"italic":true},"insert":"."},{"insert":"\nFind the\n95% HPD credible interval of\u00a0"},{"attributes":{"italic":true},"insert":"\u03b8"},{"insert":"."},{"attributes":{"list":"ordered"},"insert":"\n"},{"insert":"95% equal-tail credible interval of\u00a0"},{"attributes":{"italic":true},"insert":"\u03b8"},{"attributes":{"list":"ordered"},"insert":"\n"},{"insert":"\n"}]}

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