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A random variable x takes two values 0 and 1 with probability of x = 0is p. If x follows binomial distribution, then the mean and variance of number of 0's is given respectively by
  • a)
    np(1 - p), np
  • b)
    np, np(1 - p)
  • c)
    np, np
  • d)
    None of these
Correct answer is option 'B'. Can you explain this answer?
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A random variable x takes two values 0 and 1 with probability of x = ...
Mean and Variance of the Number of 0's in a Binomial Distribution


Introduction

In a binomial distribution, a random variable x can take two values (0 and 1) with a certain probability. The mean and variance of the number of 0's in this distribution can be calculated using the formula derived from the properties of binomial distributions.

Mean of the Number of 0's

The mean of a binomial distribution is given by the product of the number of trials (n) and the probability of success (p). In this case, the number of trials is the number of times x can take either 0 or 1, and the probability of success is the probability of x being 0.

Therefore, the mean (μ) of the number of 0's can be calculated as:
μ = n * p

Variance of the Number of 0's

The variance of a binomial distribution is given by the product of the number of trials (n), the probability of success (p), and the probability of failure (q = 1 - p). In this case, the number of trials is the same as before, and the probability of success is the probability of x being 0. The probability of failure is the complement of the probability of success.

Therefore, the variance (σ^2) of the number of 0's can be calculated as:
σ^2 = n * p * q

Conclusion

The mean and variance of the number of 0's in a binomial distribution are given by:
Mean = n * p
Variance = n * p * q

In this case, since x takes two values (0 and 1) and the probability of x being 0 is p, the mean and variance of the number of 0's is given by option (B): np and np(1 - p), respectively.
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Community Answer
A random variable x takes two values 0 and 1 with probability of x = ...
We can derive that, for binomial distribution,
P(X = r) = nCr.pr.(1 - p)n-r
Mean = np
Variance = np(1 - p)
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A random variable x takes two values 0 and 1 with probability of x = 0is p. If x follows binomial distribution, then the mean and variance of number of 0's is given respectively bya)np(1 - p), npb)np, np(1 - p)c)np, npd)None of theseCorrect answer is option 'B'. Can you explain this answer?
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