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
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
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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.
Updated On: Dec 19, 2025
- Exponential
- Bernoulli
- Binomial
- Normal
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The Correct Option is D
Solution and Explanation
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.
Therefore, the correct answer is Normal.
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