The GATE Grind

GATE 2025 DA – Question 54

Probability and Statistics · Mean, median, mode, standard deviation, correlation and covariance · 2 marks · Multiple select

Consider a coin-toss experiment where the probability of head showing up is $p$. In the $i^{th}$ coin toss, let $X_i = 1$ if head appears, and $X_i = 0$ if tail appears. Consider

$$\hat{p} = \frac{1}{n}\sum_{i=1}^{n} X_i$$

where $n$ is the total number of independent coin tosses.

Which of the following statements is/are correct?

  1. $E[\hat{p}] = p$
  2. $E[\hat{p}] = \frac{p}{n}$
  3. As $n$ increases, variance of $\hat{p}$ decreases
  4. Variance of $\hat{p}$ does not depend on $n$

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Show answer and explanation

Correct answer: (A) $E[\hat{p}] = p$; (C) As $n$ increases, variance of $\hat{p}$ decreases

Explanation

The mean is $E[\hat{p}] = \frac{1}{n}\sum E[X_i] = \frac{np}{n} = p$ (A is true, B is false). The variance is $\text{Var}(\hat{p}) = \frac{1}{n^2} \cdot np(1 - p) = \frac{p(1-p)}{n}$, which falls as $n$ grows (C is true, D is false).