Question

Can the mean of a binomial distribution be less than its variance?

Answer 1

No.

The mean of a binomial distribution is np and variance is npq.

If mean is less than its variance, then np <npq.

As both n and p are positive, we can divide both sides by np.

We get 1<q, which is not true as q <1 under all circumstances. (As p+q=1, q cannot be greater than 1)

So, the mean of a binomial distribution cannot be less than its variance.