<Ol> <Li> This definition is where X is the number k of successes given a set of r failures, and is the primary way the negative binomial distribution is defined in this article . The second alternative formula clearly shows the relationship of the negative binomial distribution to the binomial distribution . The only difference is that in the binomial coefficient of the negative binomial distribution, there are n − 1 trials to choose from (instead of n) when evaluating the number of ways that k successes can occur . This is because when you are evaluating the number of ways you can obtain k successes before you reach r failures, the last trial must be a failure . As such, the other events have one fewer positions available when counting possible orderings . </Li> <Li> The second definition is where X is the total number of n trials needed to get r failures . Since the total number of trials is equal to the number of successes plus the number of failures, the formulation is the same . The only difference in the distribution is the range is shifted by a factor of r . As such, the mean, the median, and the mode are also shifted by a factor of r . </Li> <Li> The definition where X is the number of r failures that occur for a given number of k successes . This definition is very similar to the primary definition used in this article, only that k successes and r failures are switched when considering what is being counted and what is given . Note however, that p still refers to the probability of "success". </Li> <Li> The definition where X is the number of n trials that occur for a given number of k successes . This definition is very similar to definition #2, only that k successes is given instead of r failures . Note however, that p still refers to the probability of "success". </Li> </Ol> <Li> This definition is where X is the number k of successes given a set of r failures, and is the primary way the negative binomial distribution is defined in this article . The second alternative formula clearly shows the relationship of the negative binomial distribution to the binomial distribution . The only difference is that in the binomial coefficient of the negative binomial distribution, there are n − 1 trials to choose from (instead of n) when evaluating the number of ways that k successes can occur . This is because when you are evaluating the number of ways you can obtain k successes before you reach r failures, the last trial must be a failure . As such, the other events have one fewer positions available when counting possible orderings . </Li> <Li> The second definition is where X is the total number of n trials needed to get r failures . Since the total number of trials is equal to the number of successes plus the number of failures, the formulation is the same . The only difference in the distribution is the range is shifted by a factor of r . As such, the mean, the median, and the mode are also shifted by a factor of r . </Li> <Li> The definition where X is the number of r failures that occur for a given number of k successes . This definition is very similar to the primary definition used in this article, only that k successes and r failures are switched when considering what is being counted and what is given . Note however, that p still refers to the probability of "success". </Li>

How to find mode of negative binomial distribution
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