Glossary

10.3 Geometric and negative binomial distributions

3 min read•july 19, 2024

The models the number of trials needed for the first success in independent . It's crucial for understanding scenarios with repeated attempts until a desired outcome occurs, like job interviews or manufacturing quality control.

The extends this concept, focusing on the number of failures before a specific . This makes it useful for analyzing more complex situations, such as sales calls or system reliability testing.

Geometric Distribution

Properties of geometric distribution

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  • Models number of trials needed for first success in independent Bernoulli trials
    • Bernoulli trials have two possible outcomes (success or failure)
    • pp is constant across trials
    • Trials are independent
  • (PMF): P(X=k)=(1−p)k−1pP(X = k) = (1 - p)^{k - 1}p, where kk is number of trials needed for first success
  • Mean or expected value: E(X)=1pE(X) = \frac{1}{p}
  • Variance: Var(X)=1−pp2Var(X) = \frac{1 - p}{p^2}
  • : probability of additional trials needed for success is independent of previous failed trials

Geometric distribution probability calculations

  • Calculate probability of first success on kk-th trial using PMF: P(X=k)=(1−p)k−1pP(X = k) = (1 - p)^{k - 1}p
    • If p=0.3p = 0.3, probability of first success on 4th trial is P(X=4)=(1−0.3)3⋅0.3=0.1029P(X = 4) = (1 - 0.3)^{3} \cdot 0.3 = 0.1029
  • Find probability of first success within certain number of trials by summing probabilities for each trial
    • Probability of first success within 3 trials: P(X≤3)=P(X=1)+P(X=2)+P(X=3)P(X \leq 3) = P(X = 1) + P(X = 2) + P(X = 3)
  • Calculate expected number of trials needed for first success using mean: E(X)=1pE(X) = \frac{1}{p}
    • If p=0.2p = 0.2, expected number of trials is E(X)=10.2=5E(X) = \frac{1}{0.2} = 5

Negative Binomial Distribution

Negative binomial vs geometric distributions

  • Negative binomial models number of failures before rr-th success in independent Bernoulli trials
    • rr is fixed, positive integer representing required successes
  • Success probability pp is constant across trials
  • Trials are independent
  • Geometric distribution is special case of negative binomial where r=1r = 1

Negative binomial probability calculations

  • PMF: P(X=k)=(k+r−1r−1)pr(1−p)kP(X = k) = \binom{k + r - 1}{r - 1}p^r(1 - p)^k, where kk is number of failures before rr-th success
    • If p=0.4p = 0.4 and we want probability of 3 failures before 2nd success: P(X=3)=(3+2−12−1)0.42(1−0.4)3=0.1296P(X = 3) = \binom{3 + 2 - 1}{2 - 1}0.4^2(1 - 0.4)^3 = 0.1296
  • Mean or expected value: E(X)=r(1−p)pE(X) = \frac{r(1 - p)}{p}
  • Variance: Var(X)=r(1−p)p2Var(X) = \frac{r(1 - p)}{p^2}
  • Calculate cumulative probabilities by summing individual probabilities for values less than or equal to target value

Applications of geometric and negative binomial distributions

  • Geometric distribution:
    • Model number of defective items before finding non-defective item in manufacturing
    • Determine number of job interviews needed before receiving offer
    • Analyze number of attempts before successfully completing task (free throw shots in basketball)
  • Negative binomial distribution:
    • Model number of unsuccessful attempts before achieving specified successes (sales calls to close certain number of deals)
    • Analyze failures before system experiences specified successes (password attempts before user successfully logs in certain times)
    • Determine number of inspections needed to find specified number of defective items in quality control

Key Terms to Review (18)

Bernoulli Trials: Bernoulli trials are a sequence of random experiments where each experiment has exactly two possible outcomes: success or failure. These trials are characterized by a constant probability of success, independent of other trials, which forms the foundation for understanding various probability distributions. The significance of Bernoulli trials lies in their application to model real-world scenarios, such as flipping a coin or determining whether a product is defective, leading to deeper insights into geometric and negative binomial distributions.
Failure Probability: Failure probability is the likelihood that a system, component, or process will fail to perform its intended function within a specified period or under certain conditions. This concept is crucial when evaluating risk and reliability in engineering and statistical analysis. Understanding failure probability helps in making informed decisions about system design, maintenance schedules, and resource allocation to minimize the risk of failures.
Geometric Distribution: A geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials, where each trial has two outcomes: success or failure. This distribution is characterized by its memoryless property, meaning that the probability of success remains constant across trials regardless of previous outcomes. It is particularly useful in scenarios where one seeks to determine the likelihood of the first occurrence of an event.
Independent Trials: Independent trials refer to a sequence of experiments or observations where the outcome of one trial does not influence the outcome of another. This concept is crucial in probability as it allows for the use of specific distributions, like the geometric and negative binomial distributions, to model scenarios where events are repeated until a certain condition is met, such as achieving a success or a predetermined number of successes.
K successes: In probability theory, k successes refer to the number of successful outcomes in a sequence of independent Bernoulli trials, where each trial has a fixed probability of success. This concept is essential in understanding the geometric and negative binomial distributions, as it helps describe how many trials are needed to achieve a certain number of successes and the probability of achieving those successes within a given number of trials.
Mean of Negative Binomial Distribution: The mean of a negative binomial distribution is the expected number of trials required to achieve a predetermined number of successes in a series of independent and identically distributed Bernoulli trials. This mean provides insight into the average number of attempts needed before reaching the desired success threshold, and it is calculated as $$\frac{r}{p}$$, where $$r$$ is the number of successes and $$p$$ is the probability of success in each trial. Understanding this mean helps in various applications, such as modeling wait times for events in reliability testing and understanding patterns in repeated experiments.
Memoryless Property: The memoryless property is a characteristic of certain probability distributions where the future behavior of a process does not depend on its past history. This means that the conditional probability of an event occurring in the future, given that it has not occurred up to a certain time, is the same as the unconditional probability of that event occurring from that time onward. This property is especially notable in specific distributions and processes, indicating a lack of dependence on prior outcomes.
Negative Binomial Distribution: The negative binomial distribution is a probability distribution that models the number of trials required to achieve a fixed number of successes in a sequence of independent Bernoulli trials. It is closely related to the geometric distribution, which focuses on the number of trials until the first success, while the negative binomial distribution extends this concept to multiple successes.
Number of successes: The number of successes refers to the count of favorable outcomes in a series of independent trials or experiments. This concept is central in understanding the behavior of random variables in distributions that model scenarios where events occur until a specified number of successes are achieved, particularly in situations like repeated trials in the geometric and negative binomial distributions.
Pmf formula for geometric distribution: The pmf (probability mass function) formula for geometric distribution describes the probability of experiencing the first success on the nth trial in a series of independent Bernoulli trials. It specifically quantifies the likelihood that the first success occurs after a given number of failures, making it essential for understanding scenarios where success is defined by a binary outcome (success or failure). This formula captures the relationship between the number of trials, the probability of success, and the nature of geometric distributions, which are commonly applied in various fields such as reliability testing and risk assessment.
Probability Mass Function: A probability mass function (PMF) is a function that gives the probability of a discrete random variable taking on a specific value. It assigns probabilities to each possible value in the sample space, ensuring that the sum of these probabilities equals one. The PMF helps in understanding how likely each outcome is, which is crucial when working with discrete random variables.
Queuing Theory: Queuing theory is the mathematical study of waiting lines, or queues, which helps analyze various systems where entities wait for service. This theory examines different elements like arrival rates, service rates, and the number of servers to optimize performance and efficiency. By understanding the dynamics of queues, one can apply statistical models to predict wait times and resource allocation in many real-world scenarios.
Random variable: A random variable is a numerical outcome of a random process, which can take on different values based on the result of a random event. This concept is fundamental in probability and statistics, as it allows us to quantify uncertainty and analyze various scenarios. Random variables can be classified into discrete and continuous types, helping us to connect probability distributions with real-world applications and stochastic processes.
Reliability Analysis: Reliability analysis is a statistical method used to assess the consistency and dependability of a system or component over time. It focuses on determining the probability that a system will perform its intended function without failure during a specified period under stated conditions. This concept is deeply interconnected with random variables and their distributions, as understanding the behavior of these variables is crucial for modeling the reliability of systems and processes.
Success Probability: Success probability is the likelihood of achieving a successful outcome in a single trial of a random experiment. This concept is fundamental in understanding various probability distributions, as it directly influences the behavior and characteristics of those distributions. In particular, success probability determines how likely an event is to occur and plays a critical role in modeling different types of experiments, including those involving repeated trials or specific success conditions.
Trials until success: Trials until success refers to the number of attempts or trials needed to achieve the first success in a sequence of independent Bernoulli trials, where each trial has a fixed probability of success. This concept is foundational in understanding certain types of probability distributions, which describe scenarios where outcomes are evaluated based on repeated trials until a desired result is obtained. The geometric distribution specifically models this scenario, while the negative binomial distribution extends it to situations where multiple successes are required.
Variance of Geometric Distribution: The variance of a geometric distribution measures the spread or variability of the number of trials needed to achieve the first success in a series of independent Bernoulli trials. This concept is important because it helps quantify the uncertainty involved in predicting how many attempts will be required to succeed, given a fixed probability of success on each trial. A key feature is that as the probability of success increases, the variance decreases, reflecting that successes are likely to occur sooner with higher probabilities.
Variance of Negative Binomial Distribution: The variance of a negative binomial distribution measures the spread or dispersion of the number of trials needed to achieve a fixed number of successes in a sequence of independent Bernoulli trials. It is an important characteristic that helps to understand the variability of the outcomes when the process involves repeated trials until a specified number of successes occurs. The variance is influenced by both the number of successes required and the probability of success on each trial, showcasing the distribution's behavior in various scenarios.
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