A standard approach to measure the age of or components is based on carbon datin

Question

A standard approach to measure the age of or components is based on carbon dating which measures the proportion of C^14, a radioactive isotope of carbon. The decay rate of C^14 is given by an exponential distribution. The numbers we consider in this problem are not realistic to save you from doing painful calculations. In this question you are going to try and understand the rate of decay of C^14. The unit of C time in all these questions is in 10,000 years, for ease of computation when considering a time of 10,000 years use the value t = l. A collection of initial studies on the decay time had mean mu = 2 and variance sigma^2 = 01. If we assume the distribution of decay rates are from a Gamma distribution which has density f(lambda) = beta^alpha/T(alpha) lambda^alpha - 1 e^- beta lambda, lambda > 0, where the mean and variance are alpha/beta, alpha/beta^2. Estimate the parameters alpha and beta given this information. Someone now conducts a large study where the collect many samples all of which we assume are drawn x_1,....x_n approximately^iid Exp State the two conditional densities f(x_1,, x_n | lambda), f(lambda| x_1,, x_n) as well as the joint and marginal densities f(lambda, x_1,, x_n), f(x_1,, x_n). You find out in this large study that n = 1,000 and sigma_i X_i = 2,003. State the updated estimates of the mean mu and standard deviation delta^2 of the decay parameter.

Explanation / Answer

1)

µ = / = .2
2 = /2 = .01
2 = / * 1/ = .2 * 1/ = 0.01, = .2/.01 = 20
= .2*20 = 4

Note: Please post 2nd question separately if it needs answering

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