Problem

(i) Nima wants to investigate the distribution of a discrete random variable $X$. The unknown mean of the distribution is $\mu$.

Nima takes a sample and finds the sample mean, $\bar{x}$.

For each of the following, state whether or not it is a statistic. Give a reason for your answer in each case.

(a) $\mu$

(b) $\bar{x}$

(ii) A discrete random variable $Y$ has the sampling distribution given in the table below.

$y$	2	5	6
$P(Y = y)$	$\dfrac{1}{3}$	$\dfrac{1}{4}$	$\dfrac{5}{12}$

Two random independent observations of $Y$ are taken.

Find the probability that the value of the first observation of $Y$ is smaller than the value of the second observation of $Y$.

(iii) A bag contains three counters, numbered 3, 4 and 5 respectively. Two counters are drawn at random, one after the other without replacement.

(a) List all the possible combinations of the scores on the two counters.

The random variable $T$ is defined as

$T = \text{number on the first counter} + (2 \times \text{number on the second counter})$

(b) List all the possible values of $T$

(c) Find the sampling distribution of $T$

(Total for Question 2 is 11 marks)