7. The discrete random variable \(X\) has the following probability distribution, where \(\theta\) is an unknown parameter belonging to the interval \(\left( 0 , \frac { 1 } { 3 } \right)\).
| Value of \(X\) | 1 | 3 | 5 |
| Probability | \(\theta\) | \(1 - 3 \theta\) | \(2 \theta\) |
- Obtain an expression for \(E ( X )\) in terms of \(\theta\) and show that
$$\operatorname { Var } ( X ) = 4 \theta ( 3 - \theta ) .$$
In order to estimate the value of \(\theta\), a random sample of \(n\) observations on \(X\) was obtained and \(\bar { X }\) denotes the sample mean.
- Show that
$$V = \frac { \bar { X } - 3 } { 2 }$$
is an unbiased estimator for \(\theta\).
- Find an expression for the variance of \(V\).
- Let \(Y\) denote the number of observations in the random sample that are equal to 1 .
Show that
$$W = \frac { Y } { n }$$
is an unbiased estimator for \(\theta\) and find an expression for \(\operatorname { Var } ( W )\).
- Determine which of \(V\) and \(W\) is the better estimator, explaining your method clearly.