How is the negative binomial the inverse of the binomial distribution?

In summary, the negative binomial distribution is a probability distribution that describes the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified number of failures occurs. It is the inverse of the binomial distribution, meaning it describes the number of trials needed to achieve a fixed number of successes. Yes, it can be derived from the binomial distribution by considering the number of failures instead of successes. The key differences between the two distributions are the number of trials and successes, as well as the assumption of a constant probability of success in the binomial distribution. The negative binomial distribution is commonly used in situations where the number of successes is not fixed and the probability of success may change over time, such as in
  • #1
Simfish
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Can anyone give a user-friendly explanation?

http://en.wikipedia.org/wiki/Negative_binomial_distribution#Properties

We see that the binomial distribution measures the probability of X successes after n trials, whereas the negative binomial measures the probability of the trial number after the Xth success. The question is - how does this relate to an inverse? How would the word "inverse" simplify the analogy?
 
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  • #2
It is not an inverse in the usual sense that g-1(g(x)) = x.
 
  • #3


The negative binomial distribution can be seen as the inverse of the binomial distribution in the sense that it measures the number of trials required to achieve a certain number of successes, while the binomial distribution measures the number of successes achieved after a certain number of trials. In other words, they are two sides of the same coin, with the binomial distribution focusing on the end result and the negative binomial distribution focusing on the process. Just like how the inverse of a function "undoes" the original function, the negative binomial distribution "undoes" the binomial distribution by looking at the problem from a different perspective.
 

1. What is the negative binomial distribution?

The negative binomial distribution is a probability distribution that describes the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified number of failures occurs.

2. How is the negative binomial distribution related to the binomial distribution?

The negative binomial distribution is the inverse of the binomial distribution, meaning that the binomial distribution describes the number of successes in a fixed number of trials, while the negative binomial distribution describes the number of trials needed to achieve a fixed number of successes.

3. Can the negative binomial distribution be derived from the binomial distribution?

Yes, the negative binomial distribution can be derived from the binomial distribution by considering the number of failures instead of the number of successes in a fixed number of trials.

4. What are the key differences between the negative binomial and binomial distributions?

The main differences between the negative binomial and binomial distributions are the number of trials and the number of successes. The binomial distribution has a fixed number of trials, while the negative binomial distribution has a fixed number of successes. Additionally, the binomial distribution assumes a constant probability of success, while the negative binomial distribution allows for a changing probability of success.

5. In what situations is the negative binomial distribution commonly used?

The negative binomial distribution is commonly used in situations where the number of successes is not fixed and the probability of success may change over time. It is commonly used in areas such as survival analysis, epidemiology, and quality control.

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