# Negative Binomial Distribution Calculator

## Negative Binomial Distribution Calculator

### Related Tools

Negative Binomial PMF: 0.1244

Negative Binomial CDF: 0.5801

Negative Binomial Mean: 4.5000

Negative Binomial Variance: 11.2500

Negative Binomial Standard Deviation: 3.3541

## Negative Binomial Distribution Calculator: A Comprehensive Guide to Probability, Standard Deviation, and Random Variables

A negative binomial distribution calculator is a tool used to calculate the probability of a negative binomial distribution. This type of distribution is a discrete probability distribution that describes the number of trials required to achieve a fixed number of successes. It is a generalization of the binomial distribution, which describes the probability of a fixed number of successes in a given number of trials.

The negative binomial distribution is characterized by two parameters: the number of successes required (k) and the probability of success on each trial (p). The probability mass function of the negative binomial distribution gives the probability of observing r successes before the kth success in a sequence of independent and identically distributed Bernoulli trials. The cumulative distribution function gives the probability of observing at most r successes before the kth success.

The negative binomial distribution is often used in situations where the number of trials required to achieve a fixed number of successes is random. It is also related to the geometric distribution, which describes the probability of the first success occurring on the rth trial. The variance and expected value of the negative binomial distribution can be calculated using a formula that depends on the values of k and p. A negative binomial distribution calculator can be used to quickly and easily calculate these values for a given set of parameters.

In summary, a negative binomial distribution calculator is a useful tool for calculating probabilities related to the negative binomial distribution. This distribution describes the number of trials required to achieve a fixed number of successes and is related to the binomial and geometric distributions. The calculator can be used to calculate the probability mass function, cumulative distribution function, variance, and expected value of the negative binomial distribution for a given set of parameters.

## What is Negative Binomial Distribution?

Negative binomial distribution is a type of discrete probability distribution that models the number of failures before a fixed number of successes in a series of independent and identically distributed Bernoulli trials. It is also known as the Pascal distribution or the Polya distribution.

In a negative binomial distribution, the probability of success is denoted by p, and the number of successes required is denoted by r. The random variable X represents the number of failures before the rth success occurs.

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

P(X = k) = (k + r - 1) choose (r - 1) * p^r * (1 - p)^k

where (k + r - 1) choose (r - 1) is the binomial coefficient.

The mean of the negative binomial distribution is given by:

E(X) = r * (1 - p) / p

and the variance is given by:

Var(X) = r * (1 - p) / p^2

The cumulative distribution function of the negative binomial distribution can be expressed in terms of the regularized incomplete beta function.

The negative binomial distribution is closely related to the geometric distribution, which models the number of trials needed to achieve the first success. In fact, the negative binomial distribution is sometimes referred to as the "generalized geometric distribution".

The negative binomial distribution has applications in many areas, such as reliability engineering, queuing theory, and epidemiology. For example, it can be used to model the number of defective items in a sample, or the number of infections in a population.

In summary, the negative binomial distribution is a probability distribution that models the number of failures before a fixed number of successes in a series of independent and identically distributed Bernoulli trials. It is a discrete probability distribution with possible values ranging from 0 to infinity. The probability mass function and cumulative distribution function can be calculated using the binomial coefficient and regularized incomplete beta function, respectively. The mean and variance of the distribution can also be calculated.

## How to Calculate Negative Binomial Distribution?

Calculating the negative binomial distribution requires knowledge of several key factors. These factors include the probability of success, the number of successes, the random variable, the mean, the possible values, the probability mass function, the cumulative distribution function, the variance, and the formula.

To calculate the negative binomial distribution, one must first understand the binomial distribution. The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of trials. The negative binomial distribution is a generalization of the binomial distribution that describes the number of trials required to achieve a fixed number of successes.

To calculate the negative binomial distribution, one must first determine the probability of success. The probability of success is the likelihood of achieving a successful outcome in a given trial. This value is denoted by the letter "p."

Next, one must determine the number of successes. The number of successes is the fixed number of successful outcomes that are desired. This value is denoted by the letter "k."

The random variable is the number of trials required to achieve the fixed number of successes. This value is denoted by the letter "x."

The mean is the expected number of trials required to achieve the fixed number of successes. This value is denoted by the letter "μ."

The possible values are the integer values of the random variable x. These values range from k to infinity.

The probability mass function is the function that describes the probability of each possible outcome. This function is denoted by the letter "P(x=k)."

The cumulative distribution function is the function that describes the probability of achieving k or fewer successes. This function is denoted by the letter "F(k)."

The variance is the measure of the spread of the distribution. This value is denoted by the letter "σ^2."

The formula for calculating the negative binomial distribution is:

P(x=k) = (k-1) C (r-1) * p^r * (1-p)^(k-r)

where "C" is the binomial coefficient, which is calculated as:

C(n,r) = n! / (r! * (n-r)!)

An example of using a negative binomial distribution calculator would be to calculate the probability of achieving three successes in ten trials, given a probability of success of 0.4. The calculator would use the formula and the values of k, p, and r to determine the probability of achieving three successes in ten trials.

In conclusion, calculating the negative binomial distribution requires knowledge of several key factors, including the probability of success, the number of successes, the random variable, the mean, the possible values, the probability mass function, the cumulative distribution function, the variance, and the formula. By understanding these factors and using a negative binomial distribution calculator, one can easily calculate the probability of achieving a fixed number of successes in a given number of trials.

## Negative Binomial Distribution vs. Geometric Distribution

When it comes to probability distributions, both the negative binomial distribution and the geometric distribution are commonly used in various fields. While they share some similarities, they also have some key differences. In this section, we will explore the differences between these two distributions.

### Definitions

The negative binomial distribution and geometric distribution are both discrete probability distributions. The negative binomial distribution models the number of failures that occur before a specified number of successes is reached in a series of independent and identical Bernoulli trials. The geometric distribution, on the other hand, models the number of trials that are needed to achieve the first success in a series of independent and identical Bernoulli trials.

### Notation

The notation for the negative binomial distribution is as follows:

• r: the number of successes that must be achieved
• p: the probability of success on each trial
• x: the number of failures before the rth success is achieved

The notation for the geometric distribution is as follows:

• p: the probability of success on each trial
• x: the number of trials needed to achieve the first success

### Probability Mass Function

The probability mass function (PMF) for the negative binomial distribution is:

P(X = x) = (x + r - 1) choose (r - 1) * p^r * (1 - p)^x

The PMF for the geometric distribution is:

P(X = x) = p * (1 - p)^(x-1)

### Cumulative Distribution Function

The cumulative distribution function (CDF) for the negative binomial distribution is:

F(X <= x) = I(p, r, x)

where I is the incomplete beta function.

The CDF for the geometric distribution is:

F(X <= x) = 1 - (1 - p)^x

### Differences

One key difference between the negative binomial distribution and the geometric distribution is the number of successes that must be achieved. In the negative binomial distribution, a fixed number of successes must be achieved before the experiment is considered complete. In the geometric distribution, only one success is needed.

Another difference is the number of trials that are needed. In the negative binomial distribution, the number of trials is not fixed and can vary depending on the number of failures that occur before the specified number of successes is reached. In the geometric distribution, the number of trials needed is fixed at one.

### Example

Suppose a company is conducting a survey to determine how many people out of 10 will purchase their product. If the probability of a person purchasing the product is 0.4, what is the probability that exactly 3 people will purchase the product?

Using the negative binomial distribution, we can set r = 3 and x = 7 (since 7 failures must occur before the third success is achieved). The probability of success on each trial is 0.4. Plugging these values into the PMF, we get:

P(X = 7) = (7 + 3 - 1) choose (3 - 1) * 0.4^3 * (1 - 0.4)^7 = 0.026

Using the geometric distribution, we can set x = 3. The probability of success on each trial is 0.4. Plugging these values into the PMF, we get:

P(X = 3) = 0.4 * (1 - 0.4)^2 = 0.096

As we can see, the probabilities are different for the two distributions. This is because the negative binomial distribution models the number of failures before a fixed number of successes is achieved, while the geometric distribution models the number of trials needed to achieve the first success.

In conclusion, while the negative binomial distribution and geometric distribution share some similarities, they also have some key differences. Understanding these differences can help in choosing the appropriate distribution for a given problem.

## Applications of Negative Binomial Distribution

The negative binomial distribution calculator is a useful tool for calculating the probability of a certain number of failures before a fixed number of successes are achieved in a series of independent Bernoulli trials. This distribution has various applications in probability and statistics, including:

### Pascal Distribution

The negative binomial distribution is also known as the Pascal distribution, named after the mathematician Blaise Pascal. This distribution is used to model the number of Bernoulli trials required to achieve a fixed number of successes.

### Coin Flips and Rolling Dice

The negative binomial distribution can be used to model the number of coin flips or dice rolls required to achieve a certain number of heads or a certain sum of numbers.

### Probability Mass Function and Cumulative Distribution Function

The probability mass function of the negative binomial distribution gives the probability of achieving a certain number of failures before a fixed number of successes. The cumulative distribution function gives the probability of achieving at most a certain number of failures before a fixed number of successes.

### Independent Bernoulli Trials

The negative binomial distribution assumes that the Bernoulli trials are independent, meaning that the outcome of one trial does not affect the outcome of the next.

### Count Data

The negative binomial distribution is often used to model count data, such as the number of accidents in a certain period of time or the number of defects in a certain product.

### Geometric Distribution

The negative binomial distribution is related to the geometric distribution, which models the number of Bernoulli trials required to achieve the first success.

Overall, the negative binomial distribution calculator is a valuable tool for analyzing data and making predictions based on the probability of achieving a certain number of successes or failures in a series of independent Bernoulli trials.

## Properties of Negative Binomial Distribution

The negative binomial distribution is a discrete probability distribution that describes the number of failures that occur before a fixed number of successes is reached in a series of independent trials. In this section, we will discuss some of the important properties of the negative binomial distribution.

### Mean and Variance

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

• Mean: E(X) = r(1-p)/p
• Variance: Var(X) = r(1-p)/p^2

Here, r is the number of successes, p is the probability of success, and X is the random variable that represents the number of failures before the rth success.

### Skewness and Kurtosis

The skewness and kurtosis of the negative binomial distribution are given by:

• Skewness: (2-p)/sqrt(r(1-p))
• Kurtosis: 6/r(1-p) + 3

The skewness measures the degree of asymmetry in the distribution, while the kurtosis measures the degree of peakedness or flatness.

### Gamma Function

The negative binomial distribution can be expressed in terms of the gamma function as:

• P(X=k) = (k+r-1)C(k) p^r (1-p)^k

Here, C(k) is the binomial coefficient, which represents the number of ways to choose k objects from a total of r+k-1 objects.

### Cumulants

The cumulants of the negative binomial distribution can be calculated using the formula:

• K_n = (-1)^n B_n(r,p) / n!

Here, B_n(r,p) is the nth Bell polynomial, which can be expressed in terms of the Stirling numbers of the second kind.

### Method of Solution

The negative binomial distribution can be solved using the method of generating functions. The probability generating function of the negative binomial distribution is given by:

• G(z) = (1-pz)^(-r)

Here, z is a complex variable.

### Example

Suppose a basketball player has a 70% chance of making a free throw. What is the probability that he will make his fifth free throw on his ninth attempt?

Using the negative binomial distribution, we can calculate:

• P(X=4) = (4+5-1)C(4) (0.7)^5 (0.3)^4 = 0.2001

Therefore, the probability that the basketball player will make his fifth free throw on his ninth attempt is approximately 0.2001.

In conclusion, the negative binomial distribution is a useful tool for modeling the number of failures that occur before a fixed number of successes is reached in a series of independent trials. It has several important properties, including its mean, variance, skewness, kurtosis, gamma function, cumulants, and method of solution.

## Conclusion

In conclusion, the negative binomial distribution calculator is a useful tool that can be used to calculate the probability of a certain number of failures before a given number of successes occur in a fixed number of trials. The negative binomial distribution is a discrete probability distribution that can be used to model the number of trials required to achieve a fixed number of successes.

The probability mass function and cumulative distribution function can be used to calculate the probability of obtaining a certain number of successes or failures in a given number of trials. The variance and standard deviation of the negative binomial distribution can also be calculated using the formula provided.

The negative binomial distribution is closely related to the binomial distribution and geometric distribution. The binomial distribution can be used to model the number of successes in a fixed number of trials, while the geometric distribution can be used to model the number of trials required to achieve a single success.

The normal distribution can also be used to approximate the negative binomial distribution under certain conditions. The expected value and expected number of trials can be calculated using the formula provided, and the probability density function can be used to model the distribution of the random variable x.

Overall, the negative binomial distribution calculator is a valuable tool for anyone interested in probability and statistics. By understanding the concepts and methods involved in the calculation of the negative binomial distribution, individuals can gain a deeper understanding of the underlying principles of probability and statistics.