1 The random variable \(X\) has a Cauchy distribution centred on \(m\). Its probability density function ( pdf ) is \(\mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 1 } { \pi } \frac { 1 } { 1 + ( x - m ) ^ { 2 } } , \quad \text { for } - \infty < x < \infty$$

1. Sketch the pdf. Show that the mode and median are at \(x = m\).
2. A sample of size 1 , consisting of the observation \(x _ { 1 }\), is taken from this distribution. Show that the maximum likelihood estimate (MLE) of \(m\) is \(x _ { 1 }\).
3. Now suppose that a sample of size 2 , consisting of observations \(x _ { 1 }\) and \(x _ { 2 }\), is taken from the distribution. By considering the logarithm of the likelihood function or otherwise, show that the MLE, \(\hat { m }\), satisfies the cubic equation $$\left( 2 \hat { m } - \left( x _ { 1 } + x _ { 2 } \right) \right) \left( \hat { m } ^ { 2 } - \left( x _ { 1 } + x _ { 2 } \right) \hat { m } + 1 + x _ { 1 } x _ { 2 } \right) = 0$$
4. Obtain expressions for the three roots of this equation. Show that if \(\left| x _ { 1 } - x _ { 2 } \right| < 2\) then only one root is real. How do you know, without doing further calculations, that in this case the real root will be the MLE of \(m\) ?
5. Obtain the three possible values of \(\hat { m }\) in the case \(x _ { 1 } = - 2\) and \(x _ { 2 } = 2\). Evaluate the likelihood function for each value of \(\hat { m }\) and comment on your answer.