1 The random variable $X$ has a Cauchy distribution centred on $m$. Its probability density function ( pdf ) is $\mathrm { f } ( x )$ where

$$\mathrm { f } ( x ) = \frac { 1 } { \pi } \frac { 1 } { 1 + ( x - m ) ^ { 2 } } , \quad \text { for } - \infty < x < \infty$$

(i) Sketch the pdf. Show that the mode and median are at $x = m$.\\
(ii) 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 $x _ { 1 }$.\\
(iii) 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

$$\left( 2 \hat { m } - \left( x _ { 1 } + x _ { 2 } \right) \right) \left( \hat { m } ^ { 2 } - \left( x _ { 1 } + x _ { 2 } \right) \hat { m } + 1 + x _ { 1 } x _ { 2 } \right) = 0$$

(iv) 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 $m$ ?\\
(v) 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.

\hfill \mbox{\textit{OCR MEI S4 2016 Q1 [24]}}