Discrete sample spaces: suppose there are N cable cars in San Francisco, numbered sequentially from 1 to N. You...

 

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Discrete sample spaces: suppose there are N cable cars in San Francisco, numbered sequentially from 1 to N. You see a cable car at random; it is numbered 203. You wish to estimate N. (See Goodman, 1952, for a discussion and references to several versions of this problem, and Jeffreys, 1961, Lee, 1989, and Jaynes, 2003, for Bayesian treatments.) (a) Assume your prior distribution on N is geometric with mean 100; that is, p(N) = (1/100)(99/100)N-1, for N = 1,2,.... What is your posterior distribution for N? (b) What are the posterior mean and standard deviation of N? (Sum the infinite series analytically or approximate them on the computer.) (c) Choose a reasonable 'noninformative' prior distribution for N and give the resulting posterior distribution, mean, and standard deviation for N.