Negative Binomial Distribution Calculator

Negative Binomial Distribution Formula:

\[ P(X = k) = C(k + r - 1, k) \times p^r \times (1 - p)^k \]

Number of Failures (k):

dimensionless

Number of Successes (r):

dimensionless

Success Probability (p):

0-1

Probability P(X = k):

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1. What is the Negative Binomial Distribution?

The negative binomial distribution models the number of failures (k) before achieving a specified number of successes (r) in a sequence of independent Bernoulli trials, each with the same success probability (p).

2. How Does the Calculator Work?

The calculator uses the negative binomial distribution formula:

\[ P(X = k) = C(k + r - 1, k) \times p^r \times (1 - p)^k \]

Where:

\( P(X = k) \) — Probability of k failures before r successes
\( C(k + r - 1, k) \) — Combination of (k + r - 1) choose k
\( p \) — Probability of success in each trial (0-1)
\( r \) — Number of successes required
\( k \) — Number of failures before r successes

Explanation: The formula calculates the probability of observing exactly k failures before achieving the r-th success in a sequence of independent trials.

3. Importance of Negative Binomial Distribution

Details: This distribution is widely used in various fields including quality control, epidemiology, ecology, and risk analysis. It's particularly useful for modeling over-dispersed count data where the variance exceeds the mean.

4. Using the Calculator

Tips: Enter the number of failures (k) as a non-negative integer, number of successes (r) as a positive integer, and success probability (p) as a decimal between 0 and 1 (exclusive).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between binomial and negative binomial distributions?
A: Binomial distribution counts successes in a fixed number of trials, while negative binomial distribution counts failures before a fixed number of successes.

Q2: When should I use negative binomial distribution?
A: Use it when you're interested in the number of trials needed to achieve a certain number of successes, particularly when dealing with count data that shows over-dispersion.

Q3: What are typical applications of this distribution?
A: Common applications include modeling insurance claims, biological counts, website visits, and manufacturing defect analysis.

Q4: What are the limitations of this distribution?
A: It assumes independent and identically distributed trials with constant success probability, which may not hold in all real-world scenarios.

Q5: How is this related to the geometric distribution?
A: The geometric distribution is a special case of the negative binomial distribution where r = 1 (waiting for the first success).