Mean and Variance of Binomial Distribution

Binomial distribution is a type of probability distribution that is used to model the probability of a certain number of successes in a fixed number of trials. It is defined by the following parameters:

- n: the number of trials
- p: the probability of success in each trial
- X: the number of successes in n trials

The mean and variance of a binomial distribution can be calculated using the following formulas:

- Mean = np
- Variance = np(1-p)

Comparison of Mean and Variance

The mean and variance of a binomial distribution can provide important information about the distribution. Specifically, the relationship between the mean and variance can indicate the degree of variability in the distribution.

If the variance is small relative to the mean, it suggests that the distribution is tightly clustered around the mean. Conversely, if the variance is large relative to the mean, it suggests that the distribution is spread out over a wider range of values.

With regard to the relationship between the mean and variance of a binomial distribution, there are three possible scenarios:

- Mean is greater than variance
- Mean is equal to variance
- Mean is less than variance

Mean is Greater than Variance

When the mean of a binomial distribution is greater than its variance, it suggests that the distribution is skewed towards the higher end of the range. This means that there are more successes than would be expected based on a symmetrical distribution.

In this scenario, the distribution is said to be "overdispersed." Overdispersion can occur when there is a high degree of heterogeneity in the population being sampled, or when there are other factors that cause the distribution to deviate from the expected binomial distribution.

Answer: The correct answer is option 'A' - always more than its variance.