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Official PYQ · 2021Set 2 · Q22Probability: Random Variables and DistributionsNAT1 mark

GATE CS 2021 Set 2 Q22 — Probability: Random Variables and Distributions

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For a given biased coin, the probability that the outcome of a toss is a head is 0.40.4. This coin is tossed 1,0001,000 times. Let XX denote the random variable whose value is the number of times that head appeared in these 1,0001,000 tosses. The standard deviation of XX (rounded to 22 decimal place) is _________
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Answer: 15.00 : 16.00

The number of heads XX in n=1000n = 1000 independent trials of a biased coin with probability of success p=0.4p = 0.4 follows a Binomial distribution, denoted as X∼Binomial(n,p)X \sim \text{Binomial}(n, p).

The parameters of the distribution are:

  • Number of trials, n=1000n = 1000
  • Probability of success (head), p=0.4p = 0.4
  • Probability of failure (tail), q=1−p=0.6q = 1 - p = 0.6

The variance of a Binomial random variable is given by the formula:

Var(X)=n⋅p⋅q\text{Var}(X) = n \cdot p \cdot q

Substituting the given values:

Var(X)=1000×0.4×0.6=240\text{Var}(X) = 1000 \times 0.4 \times 0.6 = 240

The standard deviation (σ\sigma) is the square root of the variance:

σ=Var(X)=240≈15.4919\sigma = \sqrt{\text{Var}(X)} = \sqrt{240} \approx 15.4919

Rounding to two decimal places, the standard deviation of XX is 15.49 (which lies within the official GATE range of 15.00 to 16.00).

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