4 A biased coin has a probability \(p\) of landing on heads, where \(0 < p < 1\) Simon spins the coin \(n\) times and the random variable \(X\) represents the number of heads. Taruni spins the coin \(m\) times, \(m \neq n\), and the random variable \(Y\) represents the number of heads. Simon and Taruni want to combine their results to find unbiased estimators of \(p\).
Simon proposes the estimator \(S = \frac { X + Y } { m + n }\) and Taruni proposes \(T = \frac { 1 } { 2 } \left[ \frac { X } { n } + \frac { Y } { m } \right]\)

1. Show that both \(S\) and \(T\) are unbiased estimators of \(p\).
2. Prove that, for all values of \(m\) and \(n , S\) is the better estimator.