3. (a) Let X₁, X2, ..., Xn be an independent and identically distributed sample from a distribution with probability den

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3 A Let X X2 Xn Be An Independent And Identically Distributed Sample From A Distribution With Probability Den 1 (291.77 KiB) Viewed 19 times

3. (a) Let X₁, X2, ..., Xn be an independent and identically distributed sample from a distribution with probability density function: f(x) = x²xe-xx Derive the maximum likelihood estimator (MLE) of X. [12 marks] (b) Independent Bernoulli trials were performed until a success was observed. Let X be the number of trials until the first success. i. What distribution does X follow? [2 marks] ii. On the 4th trial, the first success was observed. The likelihood function was constructed for this data, with a graph of it shown below. Briefly explain (1-2 sentences) what is on the x and y axes. In your own words (1-2 sentences), explain how the graph can aid finding the MLE. [8 marks] 0.0 0.2 0.4 0.6 0.8 1.0 (c) A sample of data of size 10 was collected at random from a population and the median was calculated. Sampling from the original data, with replacement, 1000 bootstrap samples were found and the median computed for each. i. If you had the vector of 1000 bootstrap medians, describe how you would use it to construct a 90% confidence interval. [7 marks] ii. Suppose a histogram was generated of the vector of bootstrap medians. Explain (2-3 sentences) what the histogram is approximating. [4 marks] (33 marks) 0.00 0.02 0.04 0.06 0.08