The NB distribution, also known as the Negative Binomial Distribution, is a probability distribution used to model the occurrence of unusual occurrences. It is a subset of the binomial distribution, a discrete probability distribution that describes the odds of a particular number of successes or failures in a defined number of independent trials.
In domains such as economics, engineering, and biology where unusual events are of interest, the Negative Binomial Distribution is frequently employed. This distribution can be used to represent the number of failures or successes up to a predetermined number of successes.
Listing of Contents:
Introduction
Knowledge of Probability Distributions
The Binomial Probability Distribution
Inverse Binomial Distribution
Distinctive features of the Negative Binomial Distribution
Negative Binomial Distribution Applications
Conclusion
Probability distributions are used to describe the likelihood that a particular event will occur. There are numerous distinct probability distributions, each with its own distinct properties and applications.
We shall examine the Negative Binomial Distribution in this essay. The distribution’s derivation from the binomial distribution, its properties, and its applications will be discussed.
Understanding Probability Distributions Prior to delving into the Negative Binomial Distribution, it is essential to have a firm grasp of the fundamentals of probability distributions.
A probability distribution is a function that describes the odds of all possibilities for a random variable. Consider the distribution to be a collection of rules that govern the probability of each conceivable outcome.
The two primary forms of probability distributions are discrete and continuous.
Probability distributions for discrete occurrences, such as tossing a coin or rolling dice, are characterised by discrete probability distributions. These distributions are characterised by a collection of alternative outcomes, each of which has a corresponding probability.
Continuous probability distributions represent the probabilities of continuous events, such as a person’s height or temperature. These distributions have an unlimited number of alternative outcomes, each with its own probability density.
The binomial distribution is a discrete probability distribution that describes the odds of a particular number of successes or failures in a fixed number of independent trials.
For instance, suppose we flip a coin ten times. The binomial distribution can be used to calculate the likelihood of receiving exactly five heads. On each given flip, the probability of obtaining heads is equal to the probability of getting tails. Using the binomial distribution, we can determine that the probability of receiving exactly five heads out of ten trials is around 0.246%.
The binomial distribution formula is as follows:
P(X = k) = (n pick k) * p^k * (1-p)^(n-k)
Where:
The probability of obtaining k successes in n trials is denoted by P(X = k).
(n choose k) represents the binomial coefficient, which equals n! / (k! * (n-k)!)
p is the probability of success on a single trial.
(1-p) represents the likelihood of failure in a single trial.
It is a subset of the binomial distribution. It is used to model the number of failures or successes until a predetermined number of successes have been achieved.
For instance, suppose we are flipping a coin and want to determine how many times we must flip it till we have three heads. The Negative Binomial Distribution can be used to compute the probability of obtaining precisely k tails prior to the appearance of the third head.
Following is the formula for the Negative Binomial Distribution:
P(X = k) = ((k+r-1) C k) * p^r * (1-p)^k
Where:
P(X = k) is the likelihood that k failures will occur before r successes.
(k+r-1) where k is the negative coefficient of the binomial distribution
p is the probability of success on a single trial.
(1-p) represents the likelihood of failure in a single trial.
r represents the amount of successes we wish to attain.
qualities of the Negative Binomial Distribution: The Negative Binomial Distribution has numerous noteworthy qualities. These consist of:
The distribution is discontinuous and any non-negative integer value is possible.
The distribution is tilted to the right, indicating that the right tail is longer than the left tail.
The distribution’s mean and standard deviation are both equal to rp / (1-p)
As p approaches 1, the distribution’s variance approaches infinity.
The Negative Binomial Distribution has numerous applications in domains such as economics, engineering, and biology. Here are few instances:
The Negative Binomial Distribution can be used to model consumer purchases in economics. For instance, a business may wish to determine the number of times a customer will purchase its product before moving to a competitor.
In engineering, the Negative Binomial Distribution can be utilised to estimate the number of component failures before a component must be repaired or replaced.
In biology, the Negative Binomial Distribution can be used to simulate the number of children generated by a species before a particular percentage of offspring reach adulthood.
In conclusion, the Negative Binomial Distribution is a helpful probability distribution for modelling uncommon events. It is a subset of the binomial distribution and can be used to describe the number of failures or successes up to a predetermined threshold.
Numerous sectors, including economics, engineering, and biology, make use of the distribution, which is characterised by its skewness, mean, and variance. Understanding how the Negative Binomial Distribution operates can aid in the modelling and analysis of rare occurrences in your subject.
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