For a given biased coin, the probability that the outcome of a toss is a head is 0.6. This coin is tossed 1,000 times. Let X denote the random variable whose value is the number of times that head appeared in these tosses. The standard deviation of X (rounded to 2 decimal places) is __________.
GATE 2021 · Engineering Mathematics · Statistics · medium
Answer: Standard deviation of X = sqrt(240) = 15.49 (rounded to 2 decimal places)
- Identify distribution: X counts heads in 1000 independent tosses; each toss has P(head) = 0.6. So X ~ Binomial(1000, 0.6).
- Compute variance: Var(X) = 1000 * 0.6 * 0.4 = 1000 * 0.24 = 240.
- Compute standard deviation: SD(X) = sqrt(240) = sqrt(4 * 60) = 2*sqrt(60) = 2 * 7.746 = 15.491...