Binomial distribution

Definition

A binomial distribution is a discrete probability distribution of the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by parameters $n$ (number of trials) and $p$ (probability of success).

5 Must Know Facts For Your Next Test

1. The mean of a binomial distribution is given by $\mu = np$.
2. The standard deviation of a binomial distribution is given by $\sigma = \sqrt{np(1-p)}$.
3. A binomial distribution can be approximated by a normal distribution if $np \geq 5$ and $n(1-p) \geq 5$.
4. The binomial coefficient, denoted as $\binom{n}{k}$ or 'n choose k', represents the number of ways to choose k successes out of n trials.
5. The probability mass function (PMF) for a binomial distribution is given by $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ where k is the number of successes.

Review Questions

• What are the two parameters that characterize a binomial distribution?
• How do you calculate the mean and standard deviation for a binomial distribution?
• Under what conditions can a binomial distribution be approximated by a normal distribution?

Related terms

Bernoulli Trial:

An experiment or process that results in a binary outcome: success or failure.

Normal Distribution: A continuous probability distribution characterized by its bell-shaped curve; used to approximate the binomial distribution under certain conditions.

Central Limit Theorem:

A statistical theory that states the sampling distribution of the sample mean will approach a normal distribution as the sample size becomes large.