Binomial Distribution

The binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent experiments, each of which yields success with probability $p$.

Key Concepts

• Binomial Experiment: A binomial experiment consists of a sequence of independent trials, each with a constant probability of success $p$.
• Success: An outcome that meets the specified criteria for the experiment.
• Failure: An outcome that does not meet the specified criteria for the experiment.
• Number of Trials: The total number of independent trials in the experiment.
• Probability of Success: The constant probability of success on each trial.

Probability Mass Function

The probability mass function (PMF) of the binomial distribution is given by:

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where:

• $X$ is the random variable counting the number of successes.
• $n$ is the number of trials.
• $p$ is the probability of success on each trial.
• $k$ is the number of successes.

Mean and Variance

The mean and variance of the binomial distribution are given by:

• Mean: $E(X) = np$
• Variance: $V(X) = np(1-p)$

Applications

The binomial distribution is used in a wide variety of applications, including:

• Quality control: To determine the probability of a certain number of defects in a manufactured product.
• Medical research: To determine the probability of a certain number of successes in a clinical trial.
• Social science: To determine the probability of a certain number of successes in a survey.
• Business: To determine the probability of a certain number of sales in a given period.

Example

Suppose we flip a coin 10 times and we are interested in the number of heads that appear. Let $X$ be the random variable counting the number of heads. Then $X$ follows a binomial distribution with parameters $n = 10$ and $p = 0.5$.

The probability of getting exactly 5 heads is given by:

$$P(X = 5) = \binom{10}{5} (0.5)^5 (0.5)^5 = 0.2461$$

The mean and variance of the distribution are:

• Mean: $E(X) = 10 \times 0.5 = 5$
• Variance: $V(X) = 10 \times 0.5 \times 0.5 = 2.5$

Binomial Distribution Formula

The binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent experiments, each of which yields success with probability $p$.

Formula

The probability of $x$ successes in $n$ independent experiments is given by the binomial distribution formula:

$$P(X = x) = \binom{n}{x}p^x(1-p)^{n-x}$$

where:

• $X$ is the random variable counting the number of successes
• $n$ is the number of experiments
• $p$ is the probability of success on each experiment
• $\binom{n}{x}$ is the binomial coefficient, which is the number of ways to choose $x$ objects from a set of $n$ objects

Example

Suppose we flip a coin 10 times and we are interested in the probability of getting exactly 5 heads. The probability of getting heads on each flip is $p = 1/2$. So, the probability of getting exactly 5 heads is:

$$P(X = 5) = \binom{10}{5}(1/2)^5(1/2)^5 = 252/1024 \approx 0.246$$

Applications

The binomial distribution is used in a variety of applications, including:

• Quality control: The binomial distribution can be used to determine the probability of a product being defective.
• Medical research: The binomial distribution can be used to determine the probability of a patient recovering from a disease.
• Social science: The binomial distribution can be used to determine the probability of a person voting for a particular candidate.

The binomial distribution is a powerful tool for modeling the probability of success in a sequence of independent experiments. It is used in a variety of applications, including quality control, medical research, and social science.

Mean and Variance of Binomial Distribution

The binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent experiments, each of which yields success with probability $p$.

Mean of Binomial Distribution

The mean of a binomial distribution is given by:

$$E(X) = np$$

where:

• $E(X)$ is the mean of the binomial distribution
• $n$ is the number of experiments
• $p$ is the probability of success on each experiment

Variance of Binomial Distribution

The variance of a binomial distribution is given by:

$$V(X) = np(1-p)$$

where:

• $V(X)$ is the variance of the binomial distribution
• $n$ is the number of experiments
• $p$ is the probability of success on each experiment

Properties of Mean and Variance of Binomial Distribution

The mean and variance of a binomial distribution have the following properties:

• The mean of a binomial distribution is always a whole number.
• The variance of a binomial distribution is always a non-negative number.
• The mean of a binomial distribution increases as $n$ increases.
• The variance of a binomial distribution increases as $n$ increases.
• The mean of a binomial distribution is equal to the product of $n$ and $p$.
• The variance of a binomial distribution is equal to the product of $n$, $p$, and $(1-p)$.

Applications of Mean and Variance of Binomial Distribution

The mean and variance of a binomial distribution are used in a variety of applications, including:

• Quality control
• Reliability engineering
• Medical research
• Social science research
• Business decision-making

The mean and variance of a binomial distribution are important measures of the central tendency and dispersion of the distribution. They can be used to make inferences about the population from which the sample was drawn.

Properties of Binomial Distribution

The binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent experiments, each of which yields success with probability $p$. It is used to model the number of successes in a sample of size $n$ drawn from a population with a success probability of $p$.

Properties of Binomial Distribution

The binomial distribution has several important properties, including:

• Binomial distribution is a discrete probability distribution. This means that it can only take on a finite number of values. In the case of the binomial distribution, the number of values it can take on is equal to the number of trials $n$.
• The mean of the binomial distribution is $np$. This means that the average number of successes in a sample of size $n$ is equal to the product of the sample size and the probability of success on each trial.
• The variance of the binomial distribution is $np(1-p)$. This means that the spread of the distribution is proportional to the product of the sample size and the probability of success on each trial.
• The binomial distribution is symmetric about the mean when $p=0.5$. This means that the distribution is evenly spread out on either side of the mean when the probability of success is equal to the probability of failure.
• The binomial distribution is a limiting case of the Poisson distribution. This means that the binomial distribution can be approximated by the Poisson distribution when the sample size is large and the probability of success is small.

Applications of Binomial Distribution

The binomial distribution is used in a wide variety of applications, including:

• Quality control: The binomial distribution can be used to determine the probability of a product being defective.
• Medical research: The binomial distribution can be used to determine the probability of a patient recovering from a disease.
• Social science research: The binomial distribution can be used to determine the probability of a person voting for a particular candidate.
• Business: The binomial distribution can be used to determine the probability of a customer making a purchase.

The binomial distribution is a powerful tool for modeling the number of successes in a sequence of independent experiments. It has a number of important properties that make it useful in a wide variety of applications.

Solved Examples on Binomial Distribution

Example 1: Coin Toss

Suppose we toss a fair coin 10 times. What is the probability of getting exactly 5 heads?

Solution:

Let $X$ be the random variable counting the number of heads in 10 tosses. Then $X$ follows a binomial distribution with parameters $n = 10$ and $p = 1/2$. The probability of getting exactly 5 heads is given by the binomial probability mass function:

$$P(X = 5) = \binom{10}{5} (1/2)^5 (1/2)^5 = 252 * (1/32) * (1/32) = 0.2461$$

Therefore, the probability of getting exactly 5 heads is 0.2461.

Example 2: Rolling a Die

Suppose we roll a fair six-sided die 6 times. What is the probability of getting exactly 3 sixes?

Solution:

Let $X$ be the random variable counting the number of sixes in 6 rolls. Then $X$ follows a binomial distribution with parameters $n = 6$ and $p = 1/6$. The probability of getting exactly 3 sixes is given by the binomial probability mass function:

$$P(X = 3) = \binom{6}{3} (1/6)^3 (5/6)^3 = 20 * (1/216) * (125/216) = 0.293$$

Therefore, the probability of getting exactly 3 sixes is 0.293.

Example 3: Quality Control

A quality control inspector randomly selects 10 items from a production line and inspects them for defects. The probability that any one item is defective is 0.05. What is the probability that exactly 2 items are defective?

Solution:

Let $X$ be the random variable counting the number of defective items in the sample of 10. Then $X$ follows a binomial distribution with parameters $n = 10$ and $p = 0.05$. The probability of getting exactly 2 defective items is given by the binomial probability mass function:

$$P(X = 2) = \binom{10}{2} (0.05)^2 (0.95)^8 = 45 * (0.0025) * (0.6634) = 0.073$$

Therefore, the probability of getting exactly 2 defective items is 0.073.

Binomial Distribution FAQs

What is the binomial distribution?

The binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of independent experiments, each of which has a constant probability of success.

What are the parameters of the binomial distribution?

The binomial distribution has two parameters:

• $n$: the number of independent experiments
• $p$: the probability of success in each experiment

What is the probability mass function of the binomial distribution?

The probability mass function of the binomial distribution is given by:

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where:

• $X$ is the random variable representing the number of successes
• $k$ is the number of successes
• $n$ is the number of independent experiments
• $p$ is the probability of success in each experiment

What is the mean of the binomial distribution?

The mean of the binomial distribution is given by:

$$E(X) = np$$

What is the variance of the binomial distribution?

The variance of the binomial distribution is given by:

$$V(X) = np(1-p)$$

What is the standard deviation of the binomial distribution?

The standard deviation of the binomial distribution is given by:

$$\sigma = \sqrt{np(1-p)}$$

What is the skewness of the binomial distribution?

The skewness of the binomial distribution is given by:

$$\gamma_1 = \frac{1-2p}{\sqrt{np(1-p)}}$$

What is the kurtosis of the binomial distribution?

The kurtosis of the binomial distribution is given by:

$$\gamma_2 = \frac{1-6p(1-p)}{np(1-p)}$$

What are some applications of the binomial distribution?

The binomial distribution is used in a variety of applications, including:

• Quality control
• Reliability engineering
• Medical research
• Social science research
• Actuarial science
• Genetics
• Finance

What are some limitations of the binomial distribution?

The binomial distribution has a number of limitations, including:

• It only applies to discrete random variables.
• It assumes that the probability of success is constant for each experiment.
• It assumes that the experiments are independent.

What are some alternatives to the binomial distribution?

There are a number of alternatives to the binomial distribution, including:

• The Poisson distribution
• The negative binomial distribution
• The geometric distribution
• The hypergeometric distribution