What is the minimum value of variance of a random variable X following binomial distribution?

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Minimum Variance of a Random Variable X following Binomial Distribution

A random variable X following binomial distribution with parameters n and p is given by:

X ~ B(n, p)

Variance of Binomial Distribution

The variance of X is given by:

Var(X) = np(1-p)

Minimum Value of Variance

For a given value of n, the minimum value of variance occurs when p = 1/2. This can be shown by differentiating the variance with respect to p:

d/dp Var(X) = n(1-2p)

Setting this derivative equal to zero and solving for p gives p = 1/2. This is a minimum because the second derivative is positive for all values of p:

d^2/dp^2 Var(X) = -2n < />

Explanation

The minimum value of variance occurs when the probability of success and failure are equal. This makes intuitive sense because if the probability of success is very high or very low, the variance will be higher because there will be more variability in the outcomes. On the other hand, if the probability of success and failure are equal, the outcomes will be more balanced, resulting in a lower variance.

Therefore, to minimize the variance of a random variable X following binomial distribution, we should choose the value of p to be 1/2.

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