Statistics 简明教程

Statistics - Negative Binomial Distribution

负二项分布是事件的独立试验序列中出现指定次数成功的之前成功和失败发生的次数的概率分布。以下是有关负二项分布实验的一些关键点。

Negative binomial distribution is a probability distribution of number of occurences of successes and failures in a sequence of independent trails before a specific number of success occurs. Following are the key points to be noted about a negative binomial experiment.

  1. The experiment should be of x repeated trials.

  2. Each trail have two possible outcome, one for success, another for failure.

  3. Probability of success is same on every trial.

  4. Output of one trial is independent of output of another trail.

  5. Experiment should be carried out until r successes are observed, where r is mentioned beforehand.

负二项分布概率可以使用以下公式计算：

Negative binomial distribution probability can be computed using following:

Formula

其中——

Where −

  1. ${x}$ = Total number of trials.

  2. ${r}$ = Number of occurences of success.

  3. ${P}$ = Probability of success on each occurence.

  4. ${1-P}$ = Probability of failure on each occurence.

  5. ${f(x; r, P)}$ = Negative binomial probability, the probability that an x-trial negative binomial experiment results in the rth success on the xth trial, when the probability of success on each trial is P.

  6. ${^{n}C_{r}}$ = Combination of n items taken r at a time.

Example

罗伯特是一名足球运动员。他射门得分率是 70%。罗伯特在第五次射门中踢进第三球的概率是多少？

Robert is a football player. His success rate of goal hitting is 70%. What is the probability that Robert hits his third goal on his fifth attempt?

Solution:

Solution:

此处成功的概率 P 为 0.70。试验的次数 x 为 5，成功的次数 r 为 3。使用负二项分布公式，计算罗伯特在第五次射门中踢进第三个球的概率。

Here probability of success, P is 0.70. Number of trials, x is 5 and number of successes, r is 3. Using negative binomial distribution formula, let’s compute the probability of hitting third goal in fifth attempt.

因此，在第五次射门中踢进第三个球的概率为 $ { 0.18522 }$.

Thus probability of hitting third goal in fifth attempt is $ { 0.18522 }$.