Distribution of order statistics

6 The continuous random variable \(Y\) has cumulative distribution function given by $$\mathrm { F } ( y ) = \begin{cases} 0 & y < a , \\ 1 - \frac { a ^ { 3 } } { y ^ { 3 } } & y \geqslant a , \end{cases}$$ where \(a\) is a positive constant. A random sample of 3 observations, \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 }\), is taken, and the smallest is denoted by \(S\).

1. Show that \(\mathrm { P } ( S > s ) = \left( \frac { a } { s } \right) ^ { 9 }\) and hence obtain the probability density function of \(S\).
2. Show that \(S\) is not an unbiased estimator of \(a\), and construct an unbiased estimator, \(T _ { 1 }\), based on \(S\). It is given that \(T _ { 2 }\), where \(T _ { 2 } = \frac { 2 } { 9 } \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } \right)\), is another unbiased estimator of \(a\).
3. Given that \(\operatorname { Var } ( Y ) = \frac { 3 } { 4 } a ^ { 2 }\) and \(\operatorname { Var } ( S ) = \frac { 9 } { 448 } a ^ { 2 }\), determine which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.
4. The values of \(Y\) for a particular sample are 12.8, 4.5 and 7.0. Find the values of \(T _ { 1 }\) and \(T _ { 2 }\) for this sample, and give a reason, unrelated to efficiency, why \(T _ { 1 }\) gives a better estimate of \(a\) than \(T _ { 2 }\) in this case.

7 A continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < a \\ 1 - \frac { a ^ { 5 } } { y ^ { 5 } } & y \geqslant a \end{array} \right.$$ where \(a\) is a parameter.
Two independent observations of \(Y\) are denoted by \(Y _ { 1 }\) and \(Y _ { 2 }\). The smaller of them is denoted by S .

1. Show that \(P ( S > \mathrm { s } ) = \frac { a ^ { 10 } } { s ^ { 10 } }\) and hence find the probability density function of \(S\).
2. Show that \(S\) is not an unbiased estimator of \(a\), and construct an unbiased estimator of \(a , T _ { 1 }\) based on \(S\).
3. Construct another unbiased estimator of \(a , T _ { 2 }\), of the form \(k \left( Y _ { 1 } + Y _ { 2 } \right)\), where \(k\) is a constant to be found.
4. Without further calculation, explain how you would decide which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.

9 The service time, \(X\) minutes, for each customer in a post office is modelled by the probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x > 0 , \\ 0 & \text { otherwise } . \end{cases}$$ Two customers begin to be served independently at the same instant. The larger of the two service times is \(T\) minutes.

1. By considering the probability that both customers have been served in less than \(t\) minutes, show that the cumulative distribution function of \(T\) is given by $$\mathrm { G } ( t ) = \begin{cases} 1 - 2 \mathrm { e } ^ { - 0.2 t } + \mathrm { e } ^ { - 0.4 t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$
2. Find the probability that both customers are served within 10 minutes.
3. Find the value of the interquartile range of \(T\).