The expected value of a negative binomial distribution, denoted as μ, represents the average number of successes required to achieve a specified number of failures, known as r. It is closely related to the probability mass function (PMF) of the distribution, which gives the probability of observing x successes after r failures. The expected value is also linked to the mean number of successes before achieving r failures, which is known as the failure rate. Furthermore, the expected value is influenced by the parameter p, which represents the probability of success on each independent trial.
Expected Value for Negative Binomial Distribution
The negative binomial distribution models the number of failures before a fixed number of successes in a sequence of independent experiments. The expected value of a negative binomial distribution is the average number of failures before the desired number of successes is achieved.
Formula:
The expected value (μ) of a negative binomial distribution with parameters r (number of successes) and p (probability of success) is given by:
μ = r / p
Interpretation:
- The expected value represents the average number of failures that will occur before achieving r successes.
- If p is high, fewer failures are expected, and the expected value decreases.
- If r is high, more failures are expected before the desired number of successes, and the expected value increases.
Properties:
- The expected value is always a positive value.
- The expected value is a linear function of r.
- The expected value is inversely proportional to p.
Example:
Suppose we have a sequence of independent experiments where the probability of success on each experiment is 0.2. If we want to observe 5 successes, the expected number of failures before achieving this goal is:
μ = 5 / 0.2 = 25
This means that, on average, we can expect to have 25 failures before we see 5 successes.
| r | p | μ |
|---|---|---|
| 3 | 0.5 | 6 |
| 6 | 0.2 | 30 |
| 10 | 0.1 | 100 |
Applications:
The negative binomial distribution is used in various applications, including:
- Modeling the number of defective items in a production process
- Analyzing the time until a desired outcome in clinical trials
- Estimating the number of accidents in a given time period
Question 1:
What does expected value represent in the context of negative binomial distribution?
Answer:
In negative binomial distribution, the expected value signifies the average number of failures required to obtain the desired number of successes. It is computed as the product of the parameters of the distribution: the number of successes (r) divided by the probability of success (p) multiplied by the failure probability (q). Therefore, the expected value is given by r/p*q.
Question 2:
How does the expected value of negative binomial distribution differ from the expected value of binomial distribution?
Answer:
Unlike binomial distribution, where the expected value measures the expected number of successes, the expected value of negative binomial distribution indicates the expected number of trials needed to attain a specified number of failures. This distinction stems from the fundamental difference between the two distributions: negative binomial distribution models the number of trials until a specified number of failures occur, while binomial distribution models the number of successes in a fixed number of trials.
Question 3:
What is the practical significance of expected value in negative binomial distribution?
Answer:
The expected value provides valuable insights for researchers and practitioners using negative binomial distribution. It serves as a benchmark against which actual observations can be compared, enabling the assessment of the likelihood of obtaining the desired number of successes within a specific number of trials. This information is crucial in areas such as quality control, where the expected number of defects or failures can be estimated to set appropriate standards and optimize production processes.
Well, there you have it, folks! We hope you’ve found this little guide on expected value for negative binomial distribution to be helpful. We know it can be a bit of a head-scratcher, but we tried to break it down in a way that’s easy to understand.