1. Let X₁,..., Xn Exponential (Ao) be iid with density given by fo(x) = Ao exp(-Xox) for x>0 (a) Write down an expression for the log-likelihood (b) Compute the score function. (c) By direct calculation show that expected value of the score function eval uated at the true parameter is zero. (d) Compute the Fisher Information for Ao by: i Computing it directly from Exo (u(XoX)²) ii Computing it using Fisher's identity (e) Suppose that you now consider the reparametrisation What is the Fisher Information for o? Ho 2n = (f) A (naughty) statistician makes a claim that they have an unbiased estima tor for the mean of this exponential and that the variance of their estimato is equal to where po is the true mean of the exponential above. Explain with justification why they are lying. h(μ) = 1/

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