Let X1, X2; …..;Xn (with n ϵ N0) be a sequence of i. i.d. Bernouilli random variables, that is, each Xi takes on the value of 1 with probability π and 0 with the complementary probability, 1-π
(a) Let X(sample mean)=(∑^n i=1(Xi)/n, what are the expected value and variance of sample mean? Explain your derivations. [5 marks]
(b) For which value of is the variance of X(bar) the largest? What is the corresponding largest value for the variance? (Hint: look at the expression for the variance of X(bar) and maximize with respect to π [20 marks]
(c) Use the largest possible variance for X(bar) to construct a 95 % confidence interval
for π.