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What is the difference between the negative binomial distribution and the binomial distribution?

I tried reading online, and I found that the negative binomial distribution is used when data points are discrete, but I think even the binomial distribution can be used for discrete data points.

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    $\begingroup$ They're both discrete. $\endgroup$
    – Glen_b
    Commented Oct 8, 2015 at 11:39
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    $\begingroup$ Simple illustration: You're selling candy door-to-door. At each door you knock on, you have probability 1/4 of selling 1 candy bar and probability 3/4 or selling 0 candy bars. Your probability of selling n bars if you knock on 50 doors is a binomial distribution in n. Your probability of having to knock on m door in order to sell 30 bars is a negative binomial distribution in m. Note that the former cuts off at 50 because you can't sell more than 50 bars, while the latter has a tail at infinity because you could just have terrible luck that day and never sell the 30th bar. $\endgroup$
    – Jerry Guern
    Commented Oct 9, 2015 at 10:32

3 Answers 3

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The difference is what we are interested in. Both distributions are built from independent Bernoulli trials with fixed probability of success, p.

With the Binomial distribution, the random variable X is the number of successes observed in n trials. Because there are a fixed number of trials, the possible values of X are 0, 1, ..., n.

With the Negative Binomial distribution, the random variable Y is the number of trials until observed the r th success is observed. In this case, we keep increasing the number of trials until we reach r successes. The possible values of Y are r, r+1, r+2, ... with no upper bound. The Negative Binomial can also be defined in terms of the number of failures until the r th success, instead of the number of trials until the r th success. Wikipedia defines the Negative Binomial distribution in this manner.

So to summarize:

Binomial:

  • Fixed number of trials (n)
  • Fixed probability of success (p)
  • Random variable is X = Number of successes.
  • Possible values are 0 ≤ X ≤ n

Negative Binomial:

  • Fixed number of successes (r)
  • Fixed probability of success (p)
  • Random variable is Y = Number of trials until the r th success.
  • Possible values are r ≤ Y

Thanks to Ben Bolker for reminding me to mention the support of the two distributions. He answered a related question here.

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    $\begingroup$ more discussion of NB here: stats.stackexchange.com/questions/6728/… . May be worth noting that binomial responses are bounded [0,N], NB responses are unbounded [0, ...] $\endgroup$
    – Ben Bolker
    Commented Oct 8, 2015 at 12:35
  • $\begingroup$ Good point, I've updated my answer to include this. $\endgroup$
    – Jelsema
    Commented Oct 8, 2015 at 15:39
  • $\begingroup$ thanks jelsema for detailed answer, i could understand it better now $\endgroup$
    – alily
    Commented Oct 14, 2015 at 7:53
  • $\begingroup$ Actually, Wikipedia defines it as the number of successes before a specified number of failures (denoted r) occurs. $\endgroup$
    – NoName
    Commented Jan 10, 2020 at 17:03
  • $\begingroup$ @NoName Further down in the article there are two alternate formulations of the Negative Binomial distribution, that's what I was referring to. $\endgroup$
    – Jelsema
    Commented Jan 18, 2020 at 13:48
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Negative binomial distribution, despite seemingly obvious relation to binomial, is actually better compared against the Poisson distribution. All three are discrete, btw.

In practical applications, NB is an alternative to Poisson when you observe the dispersion (variance) higher than expected by Poisson. Poisson is a the first choice to consider when you deal with count data, e.g. an annual number of car accident fatalities in a small town. Poisson distribution's both mean and the variance are defined by one parameter - a rate of occurrence, usually denoted as $\lambda$. As long as you estimated $\lambda$, your mean and variance follow. In fact, the mean must be equal to variance.

If your data suggests that the variance is greater than the mean (overdispersion), this rules out Poisson, then Negative binomial would be a next distribution to look at. It has more than one parameter, so its variance can be greater than the mean.

The relation of NB to binomial comes from the underlying process, as it was described in @Jelsema's answer. The process are related, so the distributions are too, but as I explained here the link to Poisson distribution is closer in practical applications.

UPDATE: Another aspect is the parameterization. Binomial distribution has two parameters: p and n. Its bona fide domain is 0 to n. In that it's not only discrete, but also defined on a finite set of numbers.

In contrast both Poisson and NB are defined on infinite set of non-negative integers. Poisson has one parameter $\lambda$, while NB has two: p and r. Note, that these two do not have parameter $n$. Thus it's one more way to see how NB and Poisson are connected.

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    $\begingroup$ I don't understand what you mean by "better compared against the Poisson distribution." The original question doesn't say what kind of modeling is desired. It doesn't even imply that one is interested in modeling at all. $\endgroup$
    – heropup
    Commented Oct 8, 2015 at 22:45
  • $\begingroup$ @heropup, OP is clearly interested in applications, and directly compares NB to Binomial. Hence, my answer is about that comparison, and that comparison to Poisson is more relevant in typical applications. $\endgroup$
    – Aksakal
    Commented Oct 9, 2015 at 13:12
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They are both discrete and represent counts when you are sampling.

Binomial distribution represents the number of successes in an experiment which its number of draws is fixed in advance ,for example suppose that three items are selected at random from a manufacturing process and each item is inspected and classified defective, $D$ , or nondefective, $N$ , we see that the sample space in this case is $S = ( DDD, DDN, DND, DNN, NDD, NDN, NND, NNN)$ .

Since Negative Binomial represents the number of failures until you draw a certain number of successes . Consider the same example and suppose the experiment is to sample items randomly until one defective item is observed. Then the sample space for this case is $S = ( D,ND,NND,NNND,... )$.

So Binomial counts successes in a fixed number of trials, while Negative binomial counts failures until a fixed number successes, but For the both we're drawing with replacement, which means that each trial has a fixed probability $p$ of success.

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