Negative binomial distribution
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Lessons
- Identifying Negative Binomial Distributions
Identify which of the following experiments below are negative binomial distributions?
i. A fair coin is flipped until head comes up 4 times. What is the probability that the coin will be flipped exactly 6 times?
ii. Cards are drawn out of a deck until 2 exactly aces are drawn. What is the probability that a total of 10 cards will be drawn?
iii. An urn contains 3 red balls and 2 black balls. If 2 balls are drawn with replacement what is the probability that 1 of them will be black?
iv. Roll a die until the first six comes up. What is the probability that this will take 3 rolls? - Determining the Negative Binomial Distribution
A fair coin is flipped until head comes up 4 times. What is the probability that the coin will be flipped exactly 6 times? - Determining the Cumulative Negative Binomial Distribution
A sculptor is making 3 exhibits for an art gallery. There is a probability of 0.75 that every piece of wood she carves into will be good enough to be part of the exhibit. What is the probability that she uses 4 pieces of wood or less?
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Topic Notes
Introduction to Negative Binomial Distribution
The negative binomial distribution is a crucial concept in probability theory and statistics. Our introduction video provides a comprehensive overview, making it an essential starting point for understanding this distribution. The negative binomial distribution is closely related to both the geometric distribution and the binomial distribution, sharing similarities and key differences with each. It models the number of successes in a sequence of independent Bernoulli trials before a specified number of failures occur. This distribution is particularly useful in various fields, including biology, finance, and quality control. Unlike the binomial distribution, which focuses on a fixed number of trials, the negative binomial distribution allows for a variable number of trials until a certain number of failures is reached. This flexibility makes it ideal for modeling scenarios where events continue until a specific condition is met, such as the number of sales calls needed to achieve a target number of successful sales.
Understanding the Negative Binomial Distribution
The negative binomial distribution is a powerful statistical concept that extends the principles of the geometric distribution to account for multiple successes. This distribution is particularly useful in modeling scenarios where we're interested in the number of trials needed to achieve a specific number of successes, rather than just one success as in the geometric distribution.
To understand the negative binomial distribution, let's first revisit its simpler counterpart, the geometric distribution. The geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. For example, it could represent the number of coin flips needed to get the first heads.
Now, imagine we want to extend this concept to multiple successes. This is where the negative binomial distribution comes into play. Instead of stopping at the first success, we continue until we reach a predetermined number of successes. This makes the negative binomial distribution more versatile and applicable to a wider range of real-world scenarios.
Let's illustrate this with a coin flipping example. Suppose we're flipping a fair coin and we want to know the probability of getting 3 heads (our successes) in 10 flips. This scenario perfectly fits the negative binomial distribution model. We're not just interested in the first heads, but in achieving a specific number of heads (3) within a certain number of trials (10).
The formula for the negative binomial distribution probability mass function is:
P(X = k) = C(k-1, r-1) * p^r * (1-p)^(k-r)
Where:
- k is the number of trials
- r is the number of successes
- p is the probability of success on each trial
- C(k-1, r-1) is the binomial coefficient, also known as "k-1 choose r-1"
Let's break down each component of this formula:
1. C(k-1, r-1): This represents the number of ways to choose r-1 successes from k-1 trials. It accounts for the different ways the successes can be arranged within the trials.
2. p^r: This term represents the probability of achieving r successes.
3. (1-p)^(k-r): This represents the probability of failing in the remaining k-r trials.
The negative binomial distribution formula essentially combines these probabilities to give us the likelihood of achieving our desired number of successes in a specific number of trials.
Comparing this to the geometric distribution, we can see that the negative binomial distribution is indeed an extension. The geometric distribution is actually a special case of the negative binomial distribution where r = 1, meaning we're only interested in the first success.
The flexibility of the negative binomial distribution makes it invaluable in various fields. In biology, it can model the distribution of parasites among hosts. In business, it can predict the number of sales calls needed to close a certain number of deals. In quality control, it can estimate the number of items that need to be inspected to find a specific number of defects.
Understanding the negative binomial distribution and its relationship to the geometric distribution provides a powerful tool for analyzing and predicting outcomes in scenarios involving multiple successes. By grasping this concept, you'll be better equipped to model and interpret a wide range of real-world phenomena that involve repeated trials and multiple desired outcomes.
Negative Binomial Distribution Formula and Its Components
The negative binomial distribution is a powerful statistical tool used to model the number of successes in a sequence of independent Bernoulli trials before a specified number of failures occur. At the heart of this distribution lies its formula, which we'll break down and explain in detail.
The negative binomial distribution formula is:
P(X = x) = C(x + r - 1, x) * p^x * (1-p)^r
Let's examine each component of this formula:
1. P(X = x): This represents the probability of achieving x successes before the r-th failure occurs.
2. C(x + r - 1, x): This is the binomial coefficient, also known as "x + r - 1 choose x". It calculates the number of ways to arrange x successes and r - 1 failures in any order.
3. p^x: This term represents the probability of x successes occurring, where p is the probability of success on each trial.
4. (1-p)^r: This term represents the probability of r failures occurring, where 1-p is the probability of failure on each trial.
Now, let's discuss the significance of the key parameters:
n (number of trials): In the negative binomial distribution, n is not fixed. The number of trials continues until a specified number of failures (r) is reached. This makes it different from the binomial distribution, where n is predetermined.
x (number of successes): This represents the number of successful outcomes before reaching the specified number of failures. It's the variable we're typically interested in predicting or analyzing.
p (probability of success): This is the probability of success on each individual trial. It remains constant throughout the sequence of trials.
r (number of failures): This is the predetermined number of failures that will end the sequence of trials. It's a fixed parameter in the negative binomial distribution.
To use the negative binomial distribution formula, follow these steps:
1. Determine the values of x, r, and p for your specific scenario.
2. Calculate the binomial coefficient C(x + r - 1, x).
3. Compute p^x and (1-p)^r.
4. Multiply all these terms together to get the final probability.
Let's walk through an example:
Suppose we're flipping a coin (p = 0.5) and want to know the probability of getting 3 heads (x = 3) before getting 2 tails (r = 2).
Step 1: We have x = 3, r = 2, and p = 0.5
Step 2: C(3 + 2 - 1, 3) = C(4, 3) = 4
Step 3: 0.5^3 = 0.125 and (1-0.5)^2 = 0.25
Step 4: 4 * 0.125 * 0.25 = 0.125
Therefore, the probability of getting 3 heads before 2 tails when flipping a fair coin is 0.125 or 12.5%.
The negative binomial distribution has numerous real-world applications. In quality control, it can model the number of items inspected before finding a certain number of defects. In epidemiology, it can represent the number of people who must be vaccinated to prevent a specific number of disease cases. In finance, it can model the number of trades before a certain number of losses occur.
Understanding the components of the negative binomial distribution formula allows for more accurate modeling of scenarios where we're interested in the number of successes before a certain number of failures. By manipulating the parameters r and p, we
Comparing Binomial and Negative Binomial Distributions
Understanding the differences and similarities between binomial distribution models and negative binomial distributions is crucial for statisticians, data scientists, and researchers across various fields. Both distributions are discrete probability distributions that deal with the number of successes in a series of independent trials, but they have distinct characteristics and applications.
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials. Each trial has only two possible outcomes: success or failure. The probability of success remains constant for each trial. Key characteristics of the binomial distribution include:
- Fixed number of trials (n)
- Constant probability of success (p) for each trial
- Independent trials
- Interest in the number of successes (X)
Negative Binomial Distribution
The negative binomial distribution, on the other hand, models the number of failures before a specified number of successes occurs. It does not have a fixed number of trials but continues until a predetermined number of successes is achieved. Key characteristics include:
- Variable number of trials
- Constant probability of success (p) for each trial
- Independent trials
- Interest in the number of failures before reaching a specified number of successes (r)
Similarities
Despite their differences, binomial and negative binomial distributions share some similarities:
- Both are discrete probability distributions
- Both involve a series of independent Bernoulli trials
- The probability of success (p) remains constant for each trial in both distributions
- Both can be used to model events with two possible outcomes
When to Use Each Distribution
The choice between binomial and negative binomial distributions depends on the nature of the problem and the information available:
- Use the binomial distribution when:
- The number of trials is fixed
- You're interested in the number of successes
- Each trial has a constant probability of success
- Use the negative binomial distribution when:
- The number of trials is not fixed
- You're interested in the number of failures before a specific number of successes
- Each trial has a constant probability of success
Real-World Applications
Binomial distribution applications:
- Quality control: Determining the number of defective items in a batch
- Medical trials: Assessing the number of patients responding to a treatment
- Marketing: Analyzing the number of successful sales calls out of a fixed number of attempts
- Elections: Predicting the number of votes a candidate might receive
Negative binomial distribution applications:
- Customer acquisition: Modeling the number of sales calls needed to acquire a certain number of customers
- Epidemiology: Studying the number of disease-free days before a specified number of infections occur
- Manufacturing: Analyzing the number of units produced before achieving a target number of high-quality items
- Sports analytics: Modeling the number of at-bats before a baseball player hits a certain number of home runs
Comparison Table
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