Q4) Bayesian inference for Gaussians (unknown mean and known variance)
A factory producing electric lamp holders performs torque tests to conform the quality of their lamp holders. They are quality tested by measuring the rotational force required to turn, open or close, a bulb on the lamp holder. Tests on a random sample of 8 lamp holders show an average torque required to be 2.5 N-m (Newton meters). Assume that the torque measurements are normally distributed with an unknown mean θ and a known standard deviation of 0.2 N-m. Suppose your prior distribution for θ is normal with mean 3 N-m and standard deviation of 1 N-m.
a) Write an expression for the posterior distribution for θ in terms of n. (Do not derive the formulae)
b) For n=100, find the mean and the standard deviation of the posterior distribution. Comment on the posterior variance.
c) Assume that the prior distribution of torque is changed, and now the prior is distributed as defined below over the range between 1 and 5:
Write a R program to implement this prior, and compute the posterior distribution
considering n = 1. Using R program find the posterior mean estimate of θ.
Sketch, on a single coordinate axes, the obtained prior, likelihood and posterior
distributions.
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