Question

If the random variable Y denotes an individual's income, Pareto's law claims that P(Y$\geq$y) = $(k/y)^{\Theta }$ , where k is the entire population's minimum income. it follow that . The income information has been collected on a random sample of n individuals: .

To answer this question enter your answer as a formula. In addition to the usual guidelines, a few more instructions for this problem: write as single variable p and $\bar{y}$ as m. These can be used as the input of functions as though they were usual variables e.g log(p), m^2, exp(m) etc... Remember p represent product of $x_{i}$ s only..not the product of any functions of them. The order statistic can be entered as min(y). Same for max(y). For all parts below, note that you will need to make your responses an expression in terms of p, m, min(y) or max(y). sum() is not a recognized function, and "x1", "x2", etc... are not recognized variables.

1) Assume k is known. Find the maximum likelihood estimator of $\Theta$

3) Assume $\Theta$ is known. Find the Maximum likelihood estimator of k.

Answer 1

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