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 \(\alpha\)

The random variable \(X\) has the discrete uniform distribution and takes the values \(\{1, \ldots, n\}\). The standard deviation of of \(X\) is \(2\sqrt{6}\). Find

1. the mean of \(X\), [3 marks]
2. P\((3 \leq X < \frac{2}{5}n)\). [3 marks]

The random variable \(X\) is equally likely to take any of the \(n\) integer values from \(m + 1\) to \(m + n\) inclusive. It is given that \(\text{E}(3X) = 30\) and \(\text{Var}(3X) = 36\). Determine the value of \(m\) and the value of \(n\). [7]

The random variable \(X\) is equally likely to take any of the \(n\) integer values from \(m + 1\) to \(m + n\) inclusive. It is given that E\((3X) = 30\) and Var\((3X) = 36\). Determine the value of \(m\) and the value of \(n\). [7]