Negative Binomial Distribution

What is Binomial Distribution?

Binomial Distribution is a probability distribution that models the number of successes in a fixed number of independent, identical trials, where each trial can result in one of two outcomes, success or failure. It is often used to analyse and predict the probability of success or failure in situations with binary outcomes, such as whether a product will be defective or not, whether a customer will make a purchase or not, or whether a project will meet a deadline or not. In case, X is distributed binomially with parameters n and p, then X ~ Bin (n,p)....

Formula of Binomial Distribution

For any variable X, binomial distribution formula can be written as:...

Properties of Binomial Distribution

Binomial distribution has a fixed number of independent trials; i.e., n.In each trial, there are only two outcomes, success or failure.The probability of success (p) remains constant across all trials.Each trial is independent, with no impact on others.It is a discrete probability distribution with specific, countable values.Probability Distribution Function (PDF) calculates probabilities for ‘x’ successes in ‘n’ trials.Mean (μ) equals np, and Variance (σ²) equals npq.The shape of the binomial curve varies based on ‘n’ and ‘p,’ tending towards symmetry with larger ‘n.’For large ‘n,’ it approximates a normal distribution (Central Limit Theorem).Cumulative Distribution Function (CDF) finds cumulative probabilities for ≤ ‘x’ successes....

Negative Binomial Distribution

In probability and statistics, the negative binomial distribution is used in tracking the number of trials required to get rth success. The PDF of negative binomial distribution is,...

Mean and Variance of Binomial Distribution

Mean (μ): The mean represents the average number of successes in a binomial distribution. It is calculated by multiplying the number of trials (n) by the probability of success (p). To find the mean, simply multiply the number of trials (how many times you are repeating the experiment) by the probability of a successful outcome in each trial. This will give the expected or average number of successes....

Shape of Binomial Distribution

Binomial Distribution may be symmetrical or skewed. If the probability of sucess, p, is equal to 0.5, then the binomial distribution would be symmetrical, regardless of the value of n. If p < 0.5, the distribution will be positively skewed; while for p > 0.5, the distribution will be negatively skewed. Further, for a given value of n, the greater is the departure from 0.5, the greater is the degree of skewness....

Solved Examples of Binomial Distribution

Example 1:...

Uses of Binomial Distribution in Business Statistics

In business, binomial distribution is used for the following reasons:...

Real-Life Scenarios of Binomial Distribution

Some real-life scenarios where binomial distribution is used are:...

Difference Between Binomial Distribution and Normal Distribution

Basis Binomial Distribution Normal Distribution Type of Distribution DiscreteContinuousEvents Finite and countableInfinite and uncountableProbability Calculation Uses specific probabilities for each outcomeUses a probability density functionShape Can be skewed or symmetric (depends on parameters)Always symmetrical (bell-shaped)Influence of Sample Size Significantly impacts the shapeLess influenced by sample sizeApproximation Approaches normal distribution with a large sample sizeAlways has a normal distribution shape...