Question

3. For this question let p = X/n, where X is a Binomial(n, p) random variable the number of successes in n independent trials, each with success probability p. i. Derive the variance of p as a function of n and p. Your proof should not rely on existing formulas for the variances of sample proportions or Binomial random variables, but it can take for granted that for independent random variables {V;}, Var(E; V:) = E, Var(V;), and that for constant a and random V, Var(aV) = a²Var(V). (Hint: Recall that a Binomial(n, p) random variable is the sum of n independent Bernoulli(p) random variables.) ?.