Glossary

4.4 Geometric Distribution

3 min read•june 25, 2024

The models the number of trials needed for the first in repeated independent experiments. It's crucial for analyzing scenarios like coin flips, product testing, or job interviews where we're interested in the attempts before a desired outcome.

Understanding the 's parameters helps interpret real-world situations. The shows the average number of trials for success, while the indicates variability. This knowledge is valuable for predicting outcomes and making informed decisions in various fields.

Geometric Distribution

Geometric distribution probability calculations

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Top images from around the web for Geometric distribution probability calculations

  • Probability a: Bernoulli trial on Vimeo View original

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  • Category:Geometric distribution - Wikimedia Commons View original

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  • Geometric distribution applies when conducting repeated , each with two possible outcomes (success or ), and the () stays constant across trials ()
    • denotes the number of trials until the first success is achieved (coin flips until heads)
    • () for the geometric distribution: P(X=[k](https://www.fiveableKeyTerm:k))=(1−p)k−1⋅pP(X = [k](https://www.fiveableKeyTerm:k)) = (1 - p)^{k - 1} \cdot p, where k is the number of trials (k = 1, 2, 3, ...)
  • Calculating probabilities using the geometric distribution formula involves:
    1. Identify the probability of success (p) for each trial (0.4 for a weighted coin)
    2. Determine the number of trials (k) for which to calculate the probability (3 flips)
    3. Substitute the values of p and k into the PMF formula and simplify (P(X=3)=(1−0.4)3−1⋅0.4=0.144P(X = 3) = (1 - 0.4)^{3 - 1} \cdot 0.4 = 0.144)

Interpreting geometric distribution parameters

  • Mean () of a geometric distribution: [E(X)](https://www.fiveableKeyTerm:E(X))=1p[E(X)](https://www.fiveableKeyTerm:E(X)) = \frac{1}{p}, where p is the probability of success on each trial
    • Represents the average number of trials needed to obtain the first success (5 dice rolls on average to get a 6)
  • Standard deviation of a geometric distribution: σ=1−pp2\sigma = \sqrt{\frac{1 - p}{p^2}}
    • Measures the variability in the number of trials needed to obtain the first success (higher standard deviation indicates more variability in the number of attempts)
  • Applying mean and standard deviation to real-world problems requires:
    • Considering the context and units of the random variable (number of job interviews until hired)
    • Explaining what the mean and standard deviation signify for the specific scenario (average and variability in the number of interviews needed)

Two cases of geometric distributions

  • : Random variable X represents the number of trials until the first success occurs
    • X can be 1, 2, 3, ..., as the first success can happen on any trial (first, second, third, etc.)
    • PMF for this case: P(X=k)=(1−p)k−1⋅pP(X = k) = (1 - p)^{k - 1} \cdot p, where k = 1, 2, 3, ...
  • : represents the number of failures before the first success occurs
    • Y can be 0, 1, 2, ..., as the first success can happen after any number of failures including 0 (no failures, one failure, two failures, etc.)
    • PMF for this case: P(Y=k)=(1−p)k⋅pP(Y = k) = (1 - p)^k \cdot p, where k = 0, 1, 2, ...
  • Key difference between the two cases:
    • Case 1 has exponent k - 1 and random variable starts at 1 (count includes the successful trial)
    • Case 2 has exponent k and random variable starts at 0 (count only includes failed trials before success)
  • The geometric distribution is a that models the number of trials needed to achieve the first success
  • (CDF) of the geometric distribution: F(X≤k)=1−(1−p)kF(X \leq k) = 1 - (1-p)^k, which gives the probability of success occurring within k or fewer trials
  • The is an extension of the geometric distribution, modeling the number of trials needed to achieve a specified number of successes

Key Terms to Review (26)

Bernoulli Trials: Bernoulli trials are a sequence of independent experiments, each with two possible outcomes, usually labeled as 'success' and 'failure'. The probability of success remains constant throughout the trials, and the trials are independent of each other. This concept is central to understanding the Geometric Distribution, which models the number of trials needed to obtain the first success in a series of Bernoulli trials.
Case 1: Case 1 refers to a specific scenario in the context of the Geometric Distribution, which is a discrete probability distribution that models the number of trials required to obtain the first success in a series of independent Bernoulli trials with a constant probability of success.
Case 2: Case 2 refers to a specific scenario in the geometric distribution where the probability of success is constant across trials, and we are interested in determining the probability of the first success occurring on the k-th trial. This case is crucial for understanding how outcomes unfold when trials are repeated until the first success is achieved, highlighting the nature of waiting time in probabilistic terms.
Central limit theorem for means: The Central Limit Theorem for Sample Means states that the distribution of sample means will approximate a normal distribution, regardless of the population's distribution, provided the sample size is sufficiently large. This approximation improves as the sample size increases.
Cumulative Distribution Function: The cumulative distribution function (CDF) is a function that describes the probability that a random variable takes on a value less than or equal to a specific value. It provides a complete picture of the distribution of probabilities for both discrete and continuous random variables, enabling comparisons and insights across different types of distributions.
Discrete Probability Distribution: A discrete probability distribution is a probability distribution that describes the probability of a random variable taking on a set of distinct, countable values. It is used to model situations where the outcomes are discrete, such as the number of successes in a fixed number of trials or the number of customers arriving at a service facility in a given time period.
Double-blind experiment: A double-blind experiment is a study where neither the participants nor the experimenters know who is receiving a particular treatment. This design helps to eliminate bias and ensure more reliable results.
E(X): E(X), also known as the expected value or mean, is a fundamental concept in probability and statistics that represents the average or central tendency of a random variable X. It provides a measure of the typical or expected outcome when the random variable is observed or an experiment is repeated many times.
Expected Value: Expected value is a fundamental concept in probability that represents the long-term average or mean of a random variable's outcomes, weighted by their probabilities. It provides a way to quantify the center of a probability distribution and is crucial in decision-making processes involving risk and uncertainty.
Failure: Failure refers to the inability to achieve a desired outcome or meet a specific goal. In the context of the Geometric Distribution, failure represents the occurrence of an event that does not meet the criteria for success, and the distribution models the number of trials required before the first successful event is observed.
Geometric distribution: A geometric distribution models the number of trials needed to get the first success in a series of independent and identically distributed Bernoulli trials. The probability of success remains constant across all trials.
Geometric Distribution: The geometric distribution is a discrete probability distribution that models the number of trials or attempts required to obtain the first success in a series of independent Bernoulli trials, where each trial has only two possible outcomes: success or failure. It represents the probability of the number of failures before the first success occurs.
Independent Trials: Independent trials refer to a series of experiments or events in which the outcome of one trial does not affect the outcome of another. This concept is crucial for understanding certain probability distributions, where the trials must be independent to apply specific mathematical models accurately. When outcomes are independent, it allows for predictable patterns and calculations regarding success and failure rates across multiple trials.
K: In probability and statistics, 'k' typically represents the number of successes in a given number of trials. This concept is crucial in various types of distributions, as it helps to determine the probability of achieving a specific number of successful outcomes. In these contexts, 'k' can vary based on the scenario being analyzed, allowing for calculations related to success rates in independent trials or draws from finite populations.
Mean: The mean, also known as the average, is a measure of central tendency that represents the arithmetic average of a set of values. It is calculated by summing up all the values in the dataset and dividing by the total number of values. The mean provides a central point that summarizes the overall distribution of the data.
Memoryless Property: The memoryless property, also known as the Markov property, is a characteristic of certain probability distributions that describes the lack of memory or dependence on past events. This property is particularly relevant in the context of the Geometric, Poisson, and Exponential distributions.
Negative Binomial Distribution: The negative binomial distribution is a discrete probability distribution that models the number of successes in a sequence of independent Bernoulli trials before a specified number of failures occur. It is a generalization of the geometric distribution, which models the number of trials until the first success occurs.
P: In statistics, 'p' typically refers to the probability of success in a given trial. It plays a crucial role in various probability distributions, influencing outcomes in scenarios involving repeated trials, such as determining the likelihood of achieving a certain number of successes. Understanding 'p' helps in forming conclusions about populations based on sample data and is key in hypothesis testing.
PMF: A Probability Mass Function (PMF) is a function that gives the probability that a discrete random variable is equal to a specific value. It essentially describes the likelihood of each possible outcome for the random variable, allowing us to understand its distribution and characteristics. PMFs are fundamental in statistics, particularly when working with discrete distributions like the geometric distribution.
Probability Mass Function: A probability mass function (PMF) is a mathematical function that gives the probability of a discrete random variable taking on a specific value. This function summarizes the distribution of probabilities for all possible outcomes, ensuring that the total probability across all values equals one. The PMF provides essential insights into the likelihood of various outcomes occurring in situations modeled by discrete distributions.
Probability of Success: The probability of success is the likelihood or chance that a particular outcome or event will occur in a given situation. It is a fundamental concept in probability theory and is especially relevant in the context of the Geometric Distribution, which models the number of trials needed to obtain the first success in a series of independent Bernoulli trials.
Random variable X: A random variable X is a numerical outcome of a random phenomenon that can take on different values based on chance. It serves as a bridge between the abstract concept of randomness and numerical analysis, allowing statisticians to quantify uncertainty and make predictions. Random variables can be classified into discrete or continuous types, which further influences their statistical behavior and the methods used to analyze them.
Random Variable Y: A random variable Y is a variable that can take on different values based on the outcome of a random experiment. It is a quantitative measure that represents the possible outcomes of a random phenomenon, allowing for the statistical analysis of uncertain events.
Sigma (Σ): Sigma (Σ) is a mathematical symbol used to represent the summation or addition of a series of numbers or values. It is a fundamental concept in statistics and is used extensively in various statistical analyses and calculations.
Standard Deviation: Standard deviation is a statistic that measures the dispersion or spread of a set of values around the mean. It helps quantify how much individual data points differ from the average, indicating the extent to which values deviate from the central tendency in a dataset.
Success: Success is the accomplishment of an aim or purpose, the attainment of popularity or profit, or the favorable or prosperous termination of attempts or endeavors. In the context of the Geometric Distribution, success refers to the occurrence of a specific event of interest in a series of independent Bernoulli trials.
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