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GATE 2021 ME (ME1) – Question 15

Engineering Mathematics · Probability and Statistics: Mean, median, mode, standard deviation, random variables and distributions · 1 mark · Multiple choice

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

  1. Exponential
  2. Bernoulli
  3. Binomial
  4. Normal

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Show answer and explanation

Correct answer: (D) Normal

Explanation

Each sample has finite mean and variance. The central limit theorem says the standardized sum approaches the normal distribution as the number of independent samples increases.

Thus **Normal (D)** is the requested approximation. Although a sum of compatible binomials is itself binomial, its large-sample approximation is normal.