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4: Discrete Random Variables

Last updated

Mar 17, 2025
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- 3.17: Solutions
- 4.0: Introduction to Discrete Random Variables

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4.0: Introduction to Discrete Random Variables
This page explains probability and discrete random variables using examples like quiz performances and phone call counts. It defines random variables, which take whole number values for experimental outcomes, and introduces probability density functions (PDFs) along with combinatorial formulas for efficient probability calculation. Key mathematical concepts, including the binomial coefficient and combinatorial formulas, are emphasized in the context of these calculations.
4.1: Hypergeometric Distribution *
This page discusses the hypergeometric probability distribution, used when probabilities change with each draw, such as drawing cards from a deck without replacement. It details the formula for calculating probabilities of specific outcomes, exemplified by drawing two aces in a poker hand. The page also outlines the conditions for applying the hypergeometric distribution and includes an example of calculating the probability of selecting gumdrops from a mix of candies.
4.2: Binomial Distribution
This page explains the binomial distribution, which models probabilities in processes with two outcomes across independent trials, characterized by a fixed number of trials and consistent success probability. It includes the binomial formula, mean and variance calculations, and discussions of approximations and Bernoulli Trials. An example illustrates a specific binomial problem involving 80 independent shots with a fixed success probability of 61.3%.
4.3: Geometric Distribution
This page explains the geometric probability distribution, highlighting its focus on trials until the first success, with characteristics like repeated Bernoulli trials. It includes examples such as calculating probabilities related to pancreatic cancer, women's literacy rates, a baseball player's batting average, and spotting Dalmatians based on specific criteria.
4.4: Poisson Distribution
This page discusses the Poisson distribution, which models the number of events in a fixed interval, applicable in fields like telecommunications and banking. It emphasizes the independence of events and known average rates. The distribution estimates probabilities for events and approximates the binomial distribution under certain conditions.
4.5: Expected Value of Discrete Random Variables **
This page covers the expected value of discrete random variables, defining it as a weighted average and long-run average. It includes examples, such as a coin toss game, and discusses expected values for common discrete distributions. The page presents theorems related to the expected values of functions of random variables, emphasizing the linearity of expected value and how to calculate it by applying functions to the random variable's values.
4.6: Variance of Discrete Random Variables
This page explains variance and standard deviation as essential features of random variables. Variance represents the average of squared deviations from the mean, while standard deviation, the square root of variance, enhances interpretation. The text includes calculations and practical examples, such as coin flips, along with important theorems about variance's non-linearity and how linear transformations impact it. It also mentions that shifting a random variable does not change its variance.
4.7: Key Terms
This page defines key statistical terms related to probability and random variables, including Bernoulli Trials, Binomial Experiments, and Binomial Probability Distribution. It explains Geometric and Hypergeometric Distributions, which differ in sampling methods, and discusses Poisson Probability Distribution. The concept of Random Variables is also highlighted, emphasizing their role in representing characteristics of populations.
4.8: Chapter Review
This page outlines key characteristics and types of probability distributions, detailing their properties and applications. It covers four specific distributions: the Hypergeometric Distribution for sampling without replacement; the Binomial Distribution for fixed independent trials; the Geometric Distribution focusing on trials until the first success; and the Poisson Distribution for modeling events over intervals.
4.9: Formula Review
This page provides an overview of four probability distributions: Hypergeometric, Binomial, Geometric, and Poisson. It includes definitions and key formulas for each, detailing the Binomial distribution's successes in independent trials, the Geometric distribution's trials until the first success, and the Poisson distribution's counting occurrences in fixed intervals, along with their mean, variance, and standard deviation relationships.
4.10: Practice
This page presents exercises on various probability distributions, including hypergeometric, binomial, geometric, and Poisson distributions, applied to scenarios like company attrition rates and muffin sales. It challenges the reader to define random variables, calculate probabilities, construct probability distributions, and determine expected values and standard deviations, all organized around specific data sets and contexts.
4.11: Homework
This page covers hypergeometric and probability distributions through various scenarios, including martial arts students, a survey on deaf births, medical advice callers, and cat wake-ups. It focuses on defining random variables, identifying distributions, and calculating probabilities and expected values related to these situations. The primary emphasis is on calculating event likelihoods and average occurrences within specific groups or timeframes.
4.12: References
This page provides a compilation of online resources and statistics on various subjects, including electricity access, distance education, NBA stats, demographics on saving and spending, and health indicators like HIV and pancreatic cancer prevalence. It also examines generational studies on Millennials, vulnerabilities in Afghanistan, motor vehicle safety, and childbirth statistics in Manila, with a focus on data from 2011 to 2013.
4.13: Solutions
This page contains statistical problems focused on analyzing random variables $X$ through probability distributions like binomial, geometric, and Poisson. It includes datasets, probability tables, and calculations for means and standard deviations, emphasizing real-world applications. The text also discusses how increasing sample sizes influence probabilities, using examples involving audited counts and cake shell pieces to illustrate statistical concepts.

Curated and edited by Kristin Kuter | Saint Mary's College, Notre Dame, IN

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