Data x1, x2... xN are drawn from the exponential distribution p(x) = 2a Find maximum likelihood estimates for the positive parameter a assuming two different priors on a: (1) a uniform prior, p(a > 0) = const; (ii) a scale-free prior, p(a > 0) oc 1/a.

For a random variable, its hazard function also referred to as the instantaneous failure rate is defined as the instantaneous risk (conditional probabilty) that an event of interest will happen in a narrow span of time duration. For a discrete random variable X, its hazard function is defined by the formula hX(k) =P(X=k+ 1|X > k) =pX(k)1−FX(k). For a Poisson distribution with λ= 4.2, find hX(k) and use R to plot the hazard function.

(b) Determine the marginal distributions of X and then of Y. (c) Find the conditional distribution of Y|X = x.

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A biased coin with probability q of landing heads is repeatedly tossed until the first head is seen. The number of tails X before the first head is modelled as a geometric distribution with probability mass function P(X = x) = g(1-q). The experiment was repeated n times and x1,x2,..., Xn tails were observed. (a) Write down the likelihood for q. Show that the maximum likelihood estimate for q is â = n n+S n where S = x₁. (b) Find the Fisher information and hence the asymptotic variance for â. (c) A Beta(ao,Bo) distribution is chosen as the prior distribution for q. Show that the posterior distribution is Beta(a₁,₁), where you should determine a₁ and ₁. (d) We have n = 5 and observed data x₁,...,x₂ = 4,2,5,6,3. (i) What is the maximum likelihood estimate â? (ii) Find an approximate 95% confidence interval for q. (iii) Before seeing the data, our probability distribution for q has mean 0.4 and standard deviation 0.2. Find values of ao and Bo corresponding to this belief. What is then the…

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