Use our Negative Binomial Distribution Calculators to calculate Negative Binomial Distribution variance, mode, median, mean, CDF, PDF, and PMF

Negative Binomial Distribution Calculator

For parameters \( r \) (number of successes) and \( p \) (probability of success), the PMF for the number of failures \( x \) (before the \( r^{\text{th}} \) success) is:

$$ P(X=x) = \binom{x + r – 1}{x}(1-p)^x\,p^r, \quad x=0,1,2,\dots $$

Step 1: Enter Parameters

Enter a positive integer (e.g., 5)

Enter a value in (0,1), e.g., 0.3

Enter a non-negative integer (e.g., 10)

Negative Binomial Distribution:
$$ P(X=x) = \binom{x+r-1}{x}(1-p)^x\,p^r, \quad x=0,1,2,\dots $$

Expected Value: $$ E[X] = \frac{r(1-p)}{p} $$    Variance: $$ \mathrm{Var}(X) = \frac{r(1-p)}{p^2} $$

Negative Binomial Distribution Calculators - Educational Guide

Negative Binomial Distribution Calculators

Welcome to our Negative Binomial Distribution Calculators! These tools are designed to help you analyze probabilities associated with the Negative Binomial distribution. Whether you're a student, researcher, or data analyst, our calculators simplify the process of performing statistical analyses related to this discrete probability model.

Table of Contents

What is the Negative Binomial Distribution?

The Negative Binomial distribution is a discrete probability distribution that models the number of failures, \( k \), observed before achieving a fixed number of successes, \( r \), in a series of independent Bernoulli trials. Each trial results in a success with probability \( p \) or a failure with probability \( 1-p \). Unlike the Binomial distribution, where the number of trials is fixed, the Negative Binomial focuses on the number of failures incurred until the target number of successes is reached.

  • \( r \): The predetermined number of successes.
  • \( p \): The probability of success on each trial, where \( 0 < p < 1 \).
  • \( k \): The number of failures before the \( r \)th success, \( k = 0, 1, 2, \dots \).
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Probability Mass Function (PMF)

For a Negative Binomial distribution with parameters \( r \) (number of successes) and \( p \) (probability of success), the PMF for the number of failures \( k \) (observed before the \( r \)th success) is given by:

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

Where:

  • \( k \): The number of failures, \( k = 0, 1, 2, \dots \).
  • \( r \): The fixed number of successes.
  • \( p \): The probability of success on each trial.
  • \( \binom{k+r-1}{k} \): A binomial coefficient representing the number of ways to arrange \( k \) failures among \( k+r-1 \) trials.
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Key Concepts

  • Discrete Outcome: The Negative Binomial distribution is defined for non-negative integers representing the count of failures.
  • Variable Number of Trials: Unlike the Binomial distribution, the total number of trials is not fixed; the process continues until \( r \) successes are achieved.
  • Constant Success Probability: The probability \( p \) remains constant for each independent trial.
  • Mean and Variance: The mean is \( \frac{r(1-p)}{p} \) and the variance is \( \frac{r(1-p)}{p^2} \), which describe the expected number and dispersion of failures.
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Step-by-Step Calculation Process

  1. Define the Parameters:

    Identify the number of successes \( r \), the success probability \( p \), and the number of failures \( k \) for which you want to calculate the probability.

  2. Compute the Binomial Coefficient:

    Calculate the binomial coefficient \( \binom{k+r-1}{k} \), which counts the number of ways to arrange \( k \) failures among \( k+r-1 \) trials.

  3. Substitute into the PMF:

    Plug the values of \( r \), \( p \), and \( k \) into the PMF formula:

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

  4. Calculate the Probability:

    Evaluate the expression to obtain the probability of observing exactly \( k \) failures before achieving \( r \) successes.

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Practical Examples

Example: Calculating Failures Before Successes

Scenario: Suppose you need to achieve \( r = 3 \) successes, and each trial has a success probability of \( p = 0.4 \). You want to calculate the probability of encountering exactly \( k = 4 \) failures before the 3rd success.

  1. Define the Parameters: Set \( r = 3 \), \( p = 0.4 \), and \( k = 4 \).
  2. Compute the Binomial Coefficient: Calculate $$\binom{4+3-1}{4} = \binom{6}{4}.$$
  3. Substitute into the PMF:

    $$P(X = 4) = \binom{6}{4} (1-0.4)^4 (0.4)^3$$

  4. Calculate the Probability:

    Evaluate the expression to obtain the probability of exactly 4 failures before achieving 3 successes.

This example shows how you can compute the likelihood of a specific number of failures occurring before the target number of successes is reached.

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Interpreting the Results

Understanding the output from the Negative Binomial Distribution Calculators is essential for accurate statistical analysis. Here's how to interpret the results:

  • PMF Value: Indicates the probability of observing exactly \( k \) failures before the \( r \)th success.
  • CDF Value: Represents the cumulative probability of encountering up to \( k \) failures before reaching \( r \) successes.
  • Mean and Variance: Provide insights into the central tendency \( \left(\frac{r(1-p)}{p}\right) \) and dispersion \( \left(\frac{r(1-p)}{p^2}\right) \) of the number of failures.

For instance, if the PMF value is calculated as 0.117 for a certain \( k \), it implies there is an 11.7% chance of encountering exactly that many failures before achieving the desired \( r \) successes.

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Applications of the Negative Binomial Distribution

The Negative Binomial distribution is widely used in various fields, including:

  • Quality Control: Estimating the number of defective items produced before achieving a set number of non-defective items.
  • Healthcare: Analyzing treatment outcomes by modeling the number of treatment failures before success.
  • Marketing: Determining the number of unsuccessful sales calls before reaching a target number of successful calls.
  • Sports Analytics: Predicting the number of failed attempts before a player achieves a set number of successful plays.
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Advantages of Using the Negative Binomial Distribution Calculators

  • Accuracy: Provides precise calculations based on established Negative Binomial distribution formulas.
  • User-Friendly: Intuitive interface suitable for users with various levels of statistical expertise.
  • Time-Efficient: Quickly compute PMF and CDF values without manual calculations.
  • Educational: Enhances understanding of discrete probability models and their practical applications.
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Conclusion

Our Negative Binomial Distribution Calculators are essential tools for anyone working with discrete probability models. By providing easy access to PMF and CDF calculations along with comprehensive educational content, these calculators support accurate and efficient statistical analyses across a variety of disciplines.

If you have any questions or need further assistance, please explore our additional resources or contact our support team.

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