I have successfully proved a) and b)i), but is stuck on b)ii). Please help!
The random variable X has a binomial distribution with parameters n and p.
(a) Show that \(\displaystyle P\, =\, \dfrac{X}{n}\) is an unbiased estimator of p.
Let \(\displaystyle U\, =\, n\, P\, (1\, -\, P).\)
(b) (i) Show that \(\displaystyle E(U)\, =\, (n\, -\, 1)\, p\, (1\, -\, p).\)
(b) (ii) Hence write down an unbiased estimator of Var(X).
The random variable X has a binomial distribution with parameters n and p.
(a) Show that \(\displaystyle P\, =\, \dfrac{X}{n}\) is an unbiased estimator of p.
Let \(\displaystyle U\, =\, n\, P\, (1\, -\, P).\)
(b) (i) Show that \(\displaystyle E(U)\, =\, (n\, -\, 1)\, p\, (1\, -\, p).\)
(b) (ii) Hence write down an unbiased estimator of Var(X).
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