Negative Binomial random variable

In summary, the conversation discusses data collection on the number of fish caught per day on a month-long fishing expedition. The data is assumed to follow a negative Binomial random variable, with parameters k and p, where E[X] represents the expected value and Var represents the variance of the variable. Before conducting a hypothesis test, estimates of k and p, denoted as K` and P`, are needed. These estimates can be obtained using the sample mean x` and sample variance s^2 of the fishing data, where k=pE[X]/(1-p) and p=E[X]/Var.
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Data is collected on the number of fish caught per day on a month long fishing expedition. It is hypothesised that the data are consistent with a negative Binomial random variable ,X , starting at 0, so that X~Neg Bin(k,p) where E[X]=k(1-p)/p and Var =k(1-p)/p^2 . However, before a hypothesis test can be performed, estimates of k and p (denoted by K`and P`, respectively) are required. Show how you can obtain K` and P` based on the sample mean x` and sample variance s^2 of the fishing data.
 
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  • #2


E[X]/Var = p and k=pE[X]/(1-p).
 

Related to Negative Binomial random variable

1. What is a Negative Binomial random variable?

A Negative Binomial random variable is a discrete probability distribution that models the number of failures that occur before a certain number of successes is reached in a series of independent trials. It is often used to model situations where the probability of success remains constant, but the number of trials needed for a certain number of successes is unknown.

2. How is a Negative Binomial random variable different from a Binomial random variable?

A Binomial random variable models the number of successes in a fixed number of trials, while a Negative Binomial random variable models the number of failures before a fixed number of successes is reached. Additionally, a Binomial random variable assumes a fixed number of trials and a fixed probability of success, while a Negative Binomial random variable allows for an unknown number of trials and a constant probability of success.

3. What are the parameters of a Negative Binomial random variable?

The parameters of a Negative Binomial random variable are the number of successes (r) and the probability of success (p). The number of trials is not considered a parameter since it is not fixed and can vary for each observation.

4. What is the formula for calculating the probability of a certain number of failures in a Negative Binomial distribution?

The formula for calculating the probability of r failures in a Negative Binomial distribution is P(x=r) = (r-1)C(r-1)(1-p)^r * p, where (r-1)C(r-1) is the combination formula and (1-p)^r * p is the probability of r failures and then one success.

5. How is a Negative Binomial random variable used in real-world applications?

A Negative Binomial random variable is commonly used in fields such as economics, biology, and social sciences to model phenomena such as the number of customers until a purchase is made, the number of infections until a vaccine is developed, or the number of accidents until a safety measure is implemented. It can also be used in quality control to determine the number of defective products in a batch.

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