CA Foundation · Quantitative Aptitude · Theoretical Distributions
For which of the following binomial distributions is the mean exactly equal to the variance, ignoring the trivial case of zero?
No binomial distribution with 0 < p < 1 has mean exactly equal to variance. The mean is np and the variance is npq, so equality requires q = 1, which means p = 0. Only the Poisson limit has equal mean and variance.
- ABinomial with n = 10 and p = 0.5
- BBinomial with n = 100 and p = 0.01 only approximately, as a limiting case
- CBinomial with n = 20 and p = 0.1
- DNo binomial distribution with 0 < p < 1 has mean exactly equal to varianceCorrect
Explanation
Binomial mean is np and variance is npq. They are equal only if q = 1, meaning p = 0, which is trivial. For 0 < p < 1, variance is always less than mean. Poisson is the limiting case where they are equal, but that is not an exact binomial.
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