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 S = sum_{i=1}^{n} X_i where n is the total number of independent coin tosses. Which of the following statements is/are correct?
A. E[p] = p
B. E[p] = P
C. As n increases, variance of p decreases
D. Variance of p does not depend on n
GATE 2025 · Engineering Mathematics · Independent Events · medium
Answer: Correct options: A and C. E[p-hat] = p (unbiased), and Var[p-hat] = p(1-p)/n which decreases as n increases.
Expectation of p-hat: E[p-hat] = E[S/n] = (1/n) * E[S] = (1/n) * np = p. So option A (E[p-hat] = p) is TRUE.
Variance of p-hat: Var[p-hat] = p(1-p)/n. This depends on n and decreases as n increases. So option C (as n increases, variance decreases) is TRUE, and option D (variance does not depend on n) is FALSE.