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GATE DA 2025 Q54: Sample proportion of heads: mean and variance

Question

Consider a coin-toss experiment where the probability of head showing up is pp. In the ithi^{\text{th}} coin toss, let Xi=1X_i = 1 if head appears, and Xi=0X_i = 0 if tail appears. Consider

p^=1n∑i=1nXi\widehat{p} = \frac{1}{n} \sum_{i=1}^{n} X_i

where nn is the total number of independent coin tosses.

Which of the following statements is/are correct?

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Select all correct options. No negative marking, no partial marks.
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Answer: A, C (official GATE 2025 final answer key)

Concept used: Mean and variance of a sample mean of independent Bernoulli variables.

First step: Use linearity: E[p^]=1n∑E[Xi]E[\widehat{p}] = \frac{1}{n}\sum E[X_i]. For the variance, independence gives Var⁡(p^)=1n2∑Var⁡(Xi)=p(1−p)n\operatorname{Var}(\widehat{p}) = \frac{1}{n^2}\sum \operatorname{Var}(X_i) = \frac{p(1 - p)}{n}.

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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