What is the variance var(X) of a binomial distribution given parameters n and p?

What is the variance var(X) of a binomial distribution given parameters n and p?

• A. var(X) = npq
• B. var(X) = n + p
• C. var(X) = n - p
• D. var(X) = p(1 - p)

Answer: (A)

The variance of a binomial distribution is given by the formula \( \text{var}(X) = npq \), where \( n \) is the number of trials, \( p \) is the probability of success on each trial, and \( q = 1 - p \) is the probability of failure. This formula arises from the properties of the binomial distribution, which describes the number of successes in \( n \) independent Bernoulli trials. For a single Bernoulli trial, the variance is \( pq \). Since the variance of independent random variables adds, the total variance for \( n \) trials is scaled by \( n \), resulting in \( \text{var}(X) = n \cdot \text{var}(Y) = n(pq) = npq \). Understanding this variance formula is crucial for calculating the spread or variability of outcomes in a binomial experiment, which helps provide insights into the likelihood of different numbers of successes occurring. Thus, the choice that states \( \text{var}(X) = npq \) accurately reflects this concept and is, therefore, the correct answer.

The variance of a binomial distribution is given by the formula ( \text{var}(X) = npq ), where ( n ) is the number of trials, ( p ) is the probability of success on each trial, and ( q = 1 - p ) is the probability of failure.

This formula arises from the properties of the binomial distribution, which describes the number of successes in ( n ) independent Bernoulli trials. For a single Bernoulli trial, the variance is ( pq ). Since the variance of independent random variables adds, the total variance for ( n ) trials is scaled by ( n ), resulting in ( \text{var}(X) = n \cdot \text{var}(Y) = n(pq) = npq ).

Understanding this variance formula is crucial for calculating the spread or variability of outcomes in a binomial experiment, which helps provide insights into the likelihood of different numbers of successes occurring. Thus, the choice that states ( \text{var}(X) = npq ) accurately reflects this concept and is, therefore, the correct answer.