TATITIC CALCULATOR
Negative Binomial Distribution
A precise tool.
What is the Negative Binomial Distribution & How does it work?
The negative binomial distribution models the probability of observing a fixed number of successes (k) before a specified number of failures (r) occurs in a series of independent Bernoulli trials with success probability (p). It is particularly useful in scenarios such as quality control, reliability testing, and over‑dispersed count data where the variance exceeds the mean. The mean of the distribution is (mu = frac{pr}{1-p}) and the variance is (sigma^2 = frac{pr}{(1-p)^2}), showing how both the success probability and the failure count shape the spread of outcomes.
P(X = k) = \binom{k+r-1}{k} (1-p)^{r} p^{k}
k = number of successes, r = number of failures, p = probability of success per trial
Parameters
Result
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Frequently Asked Questions
What is the negative binomial distribution used for?
It models scenarios like quality control, reliability testing, and over-dispersed count data where variance exceeds the mean.
How do I calculate the probability of k successes before r failures?
Use the formula P(X = k) = binom{k+r-1}{k} (1-p)^{r} p^{k}, where k is the number of successes, r is the number of failures, and p is the success probability.
Can you explain what each variable in the formula represents?
k is the number of successes, r is the number of failures, and p is the probability of success in each trial.
What are some real-world applications of the negative binomial distribution?
It's used in quality control to model the number of defective items before a certain number of good ones are found, in reliability testing for failure counts, and in analyzing over-dispersed count data.
How does the negative binomial distribution differ from the binomial distribution?
The binomial distribution models a fixed number of trials with a certain number of successes, while the negative binomial models a fixed number of successes before a certain number of failures.
What is the significance of the parameters k and r in the negative binomial distribution?
k represents the desired number of successes, and r represents the number of failures that must occur before observing those successes.
How can I interpret the result from this calculator?
The result gives you the probability of achieving exactly k successes before experiencing r failures in a series of independent trials with success probability p.
Results are for informational purposes only and do not constitute professional advice.