Find parameter from variance or other constraint

1. The random variable \(X\) has the discrete uniform distribution

$$\mathrm { P } ( X = x ) = \frac { 1 } { \alpha } \quad \text { for } x = 1,2 , \ldots , \alpha$$ The mean of a random sample of size \(n\), taken from this distribution, is denoted by \(\bar { X }\)

1. Show that \(2 \bar { X }\) is a biased estimator of \(\alpha\) A random sample of 6 observations of \(X\) is taken and the results are given below. $$\begin{array} { l l l l l l } 8 & 7 & 3 & 7 & 2 & 9 \end{array}$$
2. Use the sample mean to estimate the value of \(\alpha\)
   

2. The discrete random variable \(X\) can take any value in the set \(\{ 1,2,3,4,5,6,7,8 \}\). Arthur, Beatrice and Chris each carry out trials to investigate the distribution of \(X\). Arthur finds that \(\mathrm { P } ( X = 1 ) = 0.125\) and that \(\mathrm { E } ( X ) = 4.5\).
Beatrice finds that \(\mathrm { P } ( X = 2 ) = \mathrm { P } ( X = 3 ) = \mathrm { P } ( X = 4 ) = p\).
Chris finds that the values of \(X\) greater than 4 are all equally likely, with each having probability \(q\).

1. Calculate the values of \(p\) and \(q\).
2. Give the name for the distribution of \(X\).
3. Calculate the standard deviation of \(X\).