The variance of a binomial distribution is always given by:______________?
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Answer: A. Less than mean
Explanation
In a binomial distribution, the mean equals np and the variance equals npq. The parameter q represents the probability of failure, which must be a fraction less than one. Multiplying np by a fractional q ensures the variance is always strictly less than the mean. Binomial distributions model the number of successes in fixed independent binary trial sequences. The parameter p is success probability, and q equals one minus p. Understanding theoretical relationships between mean and variance helps identify appropriate probability models.
Quality control engineers model defective product counts using binomial variance properties. Biostatisticians analyze clinical trial success rates where binomial mean exceeds variance. Probability educators use this mathematical property to test students on discrete distribution formulas. Option B contradicts mathematical theory since variance can never exceed mean in binomial models. Option C implies Poisson-like equality, but binomial variance is reduced by the failure probability factor. Option D is imprecise because the variance is strictly less than the mean, not just less than or equal.
- Subject
- Mathematics
- Category
- Statistics
- Topic
- —
- Difficulty
- medium
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