Calculate moments from PDF

8. The random variable \(X\) has probability density function $$\begin{array} { l l } f ( x ) = 1 + \frac { 3 \lambda x } { 2 } & \text { for } - \frac { 1 } { 2 } \leqslant x \leqslant \frac { 1 } { 2 }
f ( x ) = 0 & \text { otherwise } \end{array}$$ where \(\lambda\) is an unknown parameter such that \(- 1 \leqslant \lambda \leqslant 1\).

1. 1. Find \(\mathrm { E } ( X )\) in terms of \(\lambda\).
   2. Show that \(\operatorname { Var } ( X ) = \frac { 16 - 3 \lambda ^ { 2 } } { 192 }\).
2. Show that \(\mathrm { P } ( X > 0 ) = \frac { 8 + 3 \lambda } { 16 }\). In order to estimate \(\lambda , n\) independent observations of \(X\) are made. The number of positive observations obtained is denoted by \(Y\) and the sample mean is denoted by \(\bar { X }\).
3. 1. Identify the distribution of \(Y\).
   2. Show that \(T _ { 1 }\) is an unbiased estimator for \(\lambda\), where $$T _ { 1 } = \frac { 16 Y } { 3 n } - \frac { 8 } { 3 }$$
4. 1. Show that \(\operatorname { Var } \left( T _ { 1 } \right) = \frac { 64 - 9 \lambda ^ { 2 } } { 9 n }\).
   2. Given that \(T _ { 2 }\) is also an unbiased estimator for \(\lambda\), where $$T _ { 2 } = 8 \bar { X }$$ find an expression for \(\operatorname { Var } \left( T _ { 2 } \right)\) in terms of \(\lambda\) and \(n\).
   3. Hence, giving a reason, determine which is the better estimator, \(T _ { 1 }\) or \(T _ { 2 }\). \section*{END OF PAPER}

7. The discrete random variable \(X\) has the following probability distribution, where \(\theta\) is an unknown parameter belonging to the interval \(\left( 0 , \frac { 1 } { 3 } \right)\).

1. 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.
2. 1. Show that $$V = \frac { \bar { X } - 3 } { 2 }$$ is an unbiased estimator for \(\theta\).
   2. Find an expression for the variance of \(V\).
3. 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 \(\operatorname { Var } ( W )\).
4. Determine which of \(V\) and \(W\) is the better estimator, explaining your method clearly.

7.
\includegraphics[max width=\textwidth, alt={}, center]{65369843-222f-48b2-b8cd-a1c304eac3d9-6_707_718_347_660} The diagram above shows a cyclic quadrilateral \(A B C D\), where \(\widehat { B A D } = \alpha , \widehat { B C D } = \beta\) and \(\alpha + \beta = 180 ^ { \circ }\). These angles are measured.
The random variables \(X\) and \(Y\) denote the measured values, in degrees, of \(\widehat { B A D }\) and \(\widehat { B C D }\) respectively. You are given that \(X\) and \(Y\) are independently normally distributed with standard deviation \(\sigma\) and means \(\alpha\) and \(\beta\) respectively.

1. Calculate, correct to two decimal places, the probability that \(X + Y\) will differ from \(180 ^ { \circ }\) by less than \(\sigma\).
2. Show that \(T _ { 1 } = 45 ^ { \circ } + \frac { 1 } { 4 } ( 3 X - Y )\) is an unbiased estimator for \(\alpha\) and verify that it is a better estimator than \(X\) for \(\alpha\).
3. Now consider \(T _ { 2 } = \lambda X + ( 1 - \lambda ) \left( 180 ^ { \circ } - Y \right)\).
   1. Show that \(T _ { 2 }\) is an unbiased estimator for \(\alpha\) for all values of \(\lambda\).
   2. Find \(\operatorname { Var } \left( T _ { 2 } \right)\) in terms of \(\lambda\) and \(\sigma\).
   3. Hence determine the value of \(\lambda\) which gives the best unbiased estimator for \(\alpha\).