Question 1 (20 points in total). The following data are assumed to be an i.i.d. sample from a

Cauchy(,1) distribution:

1.77, -0.23, 2.76, 3.80, 3.47, 56.75, -1.34, 4.24, -2.44, 3.29, 3.71, -2.40, 4.53, -0.07, -1.05, -13.87,

-2.53, -1.75, 0.27, 43.21

The density function of the Cauchy(,1) distribution is

f(x) = (1/)[(x)^2 +1]^1

(a) (2 points) Draw a dot plot of this dataset.

(b) (4 points) Plot the log likelihood function and the first derivative of the log likelihood function.

(c) (14 points) Find the MLE for using the Newton-Raphson method. Try with all of the

following starting points: -11, -1, 0, 1.5, 4, 4.7, 7, 8, and 38. Discuss your results. Is the mean

of the data a good starting point?