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/

Let X ~ Uniform(a, b). Find Expected value EX.

Assume the time required to fully recharge an electric car is uniformly distributed between 160 and 220 minutes.  What is the expected charge time, E(x)?

the probability density function for the waiting time was given by f(x) = 0 for x 15 a) Find the expected value of the waiting time, X minutes. b) Find the variance and standard deviation of the waiting time. c) What is the probability that the waiting time is within two standard devia- tions of its expected mean value? 1 15