## Binomial Distribution

Binomial distribution is a statistical distribution for discrete data that take only a finite number of plausible values (e.g., 0, 1, 2, ..., n). It was originated from modeling the total number of Heads from n independent tosses of a coin. If the probability of getting a Head is p, the probability of getting a total of k Heads is

Then, we say the number of Heads follows a Binomial distribution with the parameters n and p. Note the mean and variance of the Binomial distribution are np and np(1-p), respectively.

**Example 1:**

When examining the prevalence of people with high blood pressure, the number of subjects in the sample follows a Binomial distribution provided the following conditions hold:

for a subject in the sample, whether the subject has high blood pressure would not influence the chance of having high blood pressure in all other subjects, and

all subjects bear the same chance of having high blood pressure.

**Example 2:**

To examine the effectiveness of a vaccine in preventing a virus infection, a number of vaccinated subjects are observed for virus infection. Then, the number of vaccinated subjects who would later be infected by the virus follows a Binomial distribution provided the following conditions hold:

Whether a vaccinated subject is infected would not influence the chance of infection in all other subjects, and

all subjects bear the same chance of infection.