2. By applying the central limit theorem to a sequence of random variables with the Bernoulli distribution, or otherwise, prove the following result in analysis. If 0 < p = 1 − q < 1 and x > 0, then X

n k



pkqn−k →2 Z x 0

1

√2π

e−1 2 u2 du as n → ∞, where the summation is over all values of k satisfying np − x√npq ≤ k ≤ np + x√npq