Question:

The maximum value of the variance of a Binomial distribution with parameters \( n \) and \( p \) is:

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Variance Optimization}
Use derivative or symmetry to maximize \( p(1 - p) \)
Maximum at \( p = \frac{1}{2} \)
Plug into variance formula: \( np(1 - p) \)
Updated On: May 19, 2025
  • \( \frac{n}{2} \)
  • \( \frac{n}{4} \)
  • \( np(1 - p) \)
  • \( 2n \)
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The Correct Option is B

Solution and Explanation

Variance of binomial distribution: \[ \text{Var} = np(1 - p) \] Maximum value of \( p(1 - p) \) occurs at \( p = \frac{1}{2} \): \[ p(1 - p) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \Rightarrow \text{Maximum variance} = n \cdot \frac{1}{4} = \frac{n}{4} \]
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