Final answer:
The central limit theorem applies to the distributions of sums and means of independent random variables from any distribution, including those that are not normally distributed such as the negative binomial distribution, given that the sample size is large enough.
Step-by-step explanation:
True. The central limit theorem (CLT) states that for a large sample size, the sampling distribution of the sample means approaches a normal distribution, regardless of the shape of the original distribution. This holds true even when samples are drawn from populations that are not normally distributed, such as the negative binomial distribution.
When dealing with independent but not identically distributed random variables, like different negative binomial distributions, the CLT can still apply, but with some conditions. As sample sizes increase, the distribution of their sums or averages tends toward the normal distribution. This is crucial for practical applications because it enables the use of normal distribution techniques even when the original data doesn't follow a normal distribution.
The theorem is robust in that it only requires the random variables to be independent. However, extremely large sample sizes may be needed if the distributions of the random variables differ significantly from each other. As a special note, when specifically talking about the negative binomial distribution, the CLT provides a basis for using normal approximations for calculating probabilities when dealing with a large number of trials.
In summary, the central limit theorem applies to the sums and means of random variables from any distribution, including the negative binomial distribution, provided the samples are independent and the sample size is sufficiently large.