The problem implies Xn is a binomial random variable, Xn Bin(n, p_n), as this is the context for such limit theorems. We are given two conditions:

You're on a roll — Step 1: Analyze the given conditions for the random variable X_n. The problem implies X_n is a binomial random variable, X_n Bin(n, p_n), as this is the context for such limit theorems. We are given two conditions: 1. p_n 0 as n . This means the probability of success for each trial becomes very small. 2. np_n as n . This means the expected number of successes grows infinitely large. Step 2: Evaluate the applicability of different asymptotic approximations. A. Binomial: X_n is already a binomial random variable. The question asks for an approximation*, implying a different distribution that X_n converges to under these conditions, not its original form. B. Poisson: The Poisson approximation to the binomial distribution is valid when n , p_n 0, and np_n , where is a finite positive constant*. In this problem, np_n , so the Poisson approximation is not appropriate. C. Bernoulli: A Bernoulli distribution describes a single trial (n=1). It is not an asymptotic approximation for a sequence of n trials as n . D. Normal: The Normal approximation to the binomial distribution (also known as the De Moivre-Laplace theorem, a special case of the Central Limit Theorem) states that if X_n Bin(n, p_n), then for large n, X_n can be approximated by a normal distribution N(, ^2) where = np_n and ^2 = np_n(1-p_n). The conditions for this approximation to be good are typically that both np_n and n(1-p_n) are sufficiently large (e.g., 5 or 10). Given np_n , the mean is growing large. Given p_n 0, it implies 1-p_n 1. Since np_n and p_n 0, n must also tend to infinity. Therefore, n(1-p_n) ≈ n . Since both np_n and n(1-p_n) , the conditions for the normal approximation are met. The standardized variable (X_n - np_n)/(sqrt(np_n(1-p_n))) converges in distribution to a standard normal distribution. Step 3: Conclude the best approximation. Based on the conditions p_n 0 and np_n , the binomial distribution X_n is best approximated by a Normal distribution. The final answer is D Got more? Send 'em.