Question

Consider a binomial random variable \( X \). If \( X_1, X_2, \dots, X_n \) are independent and identically distributed samples from the distribution of \( X \) with sum \( Y = \sum_{i=1}^{n} X_i \), then the distribution of \( Y \) as \( n \to \infty \) can be approximated as

• A. Exponential
• B. Bernoulli
• C. Binomial
• D. Normal

Hint:

The Central Limit Theorem states that the sum of a large number of i.i.d. random variables, regardless of the original distribution, will approximate a normal distribution.

Answer 1

The Correct Option is D

The sum of a large number of independent, identically distributed (i.i.d.) random variables follows a Normal distribution as per the Central Limit Theorem. Since \( Y = \sum_{i=1}^{n} X_i \) is the sum of \( n \) independent binomial random variables, the distribution of \( Y \) will approximate a normal distribution as \( n \to \infty \).
Therefore, the correct answer is Normal.