Understanding and Proving the Relationship Between Mean and Variance in Binomial Distribution

Binomial distribution is a fundamental concept in probability theory, widely used in various fields such as statistics, finance, and computer science. A binomial distribution describes the probability of achieving a certain number of successes, r, in n independent Bernoulli trials, each with the same probability of success, p. In this article, we will explore the properties of a binomial distribution, particularly focusing on the relationship between its mean and variance.

Mean and Variance of Binomial Distribution

A binomial distribution is characterized by two parameters: the number of trials, n, and the probability of success in each trial, p. The mean, E(X), and variance, Var(X), of a binomial distribution can be calculated using the following formulas:

E(X) np Var(X) np(1 - p)

Here, 1 - p is denoted as q, where q represents the probability of failure.

Proving the Mean is Greater Than or Equal to the Variance

To prove that the mean of a binomial distribution is always greater than or equal to the variance, we start by expressing the mean and variance:

E(X) np

Var(X) npq np(1 - p)

We can now calculate the mean minus the variance:

E(X) - Var(X) np - npq

Substituting q 1 - p into the equation, we get:

np - np(1 - p) np - np np^2 np^2

Since 0 ≤ p ≤ 1, it follows that 0 ≤ p^2 ≤ 1. Therefore, np^2 ≥ 0.

Thus, E(X) - Var(X) ≥ 0, which means:

E(X) ≥ Var(X)

Ratio of Variance to Mean

We can also evaluate the ratio of the variance to the mean:

Var(X) / E(X) npq / np q

Given that q 1 - p, we have:

Var(X) / E(X) 1 - p ≤ 1

This ratio implies that the variance is less than or equal to the mean, or the mean is greater than or equal to the variance, as per the definition of q.

Conclusion

For a binomial distribution defined by parameters n and p, the mean is always greater than or equal to the variance. This relationship can be mathematically proven by examining the formulas for the mean and variance, and understanding the implications of the probability of success and failure. By substituting the values and ensuring the conditions of the probability are met, we can validate the claim.

For instance, consider the following scenario:

n 10 p 0.6

The mean E(X) is:

E(X) 10 * 0.6 6

The variance Var(X) is:

Var(X) 10 * 0.6 * 0.4 2.4

Clearly, E(X) 6 ≥ 2.4 Var(X).

In conclusion, the mean of a binomial distribution is always greater than or equal to its variance, reflecting the inherent properties of the distribution and its parameters.