7. A bag contains marbles of which an unknown proportion \(p\) is red. A random sample of \(n\) marbles is drawn, with replacement, from the bag. The number \(X\) of red marbles drawn is noted. A second random sample of \(m\) marbles is drawn, with replacement. The number \(Y\) of red marbles drawn is noted. Given that \(p _ { 1 } = \frac { a X } { n } + \frac { b Y } { m }\) is an unbiased estimator of \(p\),

1. show that \(a + b = 1\). Given that \(p _ { 2 } = \frac { ( X + Y ) } { n + m }\),
2. show that \(p _ { 2 }\) is an unbiased estimator for \(p\).
3. Show that the variance of \(p _ { 1 }\) is \(p ( 1 - p ) \left( \frac { a ^ { 2 } } { n } + \frac { b ^ { 2 } } { m } \right)\).
4. Find the variance of \(p _ { 2 }\).
5. Given that \(a = 0.4 , m = 10\) and \(n = 20\), explain which estimator \(p _ { 1 }\) or \(p _ { 2 }\) you should use.