In this assignment, you will provide a plain language explanation of the primary concepts of this topic and the types of probability distributions that may be used for it. In addition, you will provide examples of how your field would apply each of these probability calculations in your field to predict or present issues in and around the pandemic.

QUESTION

Purpose

In this assignment, you will provide a plain language explanation of the primary concepts of this topic and the types of probability distributions that may be used for it. In addition, you will provide examples of how your field would apply each of these probability calculations in your field to predict or present issues in and around the pandemic.

Assignment Preparation

• Review the key concepts in Chapter 3 (in blue font in the text, the things other than article citations) – I have resources that you can link to from the Communication Instructors Resources for Understanding Probability and Statistics page, that can be helpful in understanding the key concepts before getting into the numbers and calculations.

Assignment Requirements

While the primary purpose of this assignment is for you to think about and work with statistical concepts, it is important after you complete your writing that you take the time to review and edit it for coherence, completeness, and grammar. Using a grammar and spelling program such as Grammarly is highly recommended, as excessive grammar errors and incoherent writing will lower your grade.

Assignment

1. Use ONLY words (no formulas, no equations, and no symbols): Define in your own words what a random variable is and what makes it discrete? What limitations are there in using this approach to probability prediction? What type of data do you need?
2. Define Expected Value in words only.
3. For the following types of distributions explain in words (no formulas, no equations, and no symbols) what they model and what the given elements are for each (i.e. event, sample, conditional/independent). Then provide one example from your field *of how they could be applied (you may use sources but must have more than one, you may not quote them, and you must cite them):
   • Binomial
   • Hypergeometric
   • Negative Binomial
4. How does the Bernoulli random variable relate to the binomial and negative binomial distributions?

* Think of the applications in your field as starting points for your group project case research. You may start developing that as early as you like. However, each group member must have a different case to present.

Assignment Requirements – Part B

If some, but not all of your group members post part A, you are not responsible for responding to assignments posted within the 24 hours before part B is due. If no group members post part A, everyone will get no points. If you alone post, you alone get points.

• Review group members submittals and briefly respond
  • To each of the key elements of statistics that were identified to address if the elements identified correctly, and why you think so or not. Complete this by cutting and pasting the elements list into your response and responding to each.
  • To the concerns or strengths, your group member provided on their article. This may be written as a bulleted list OR a short paragraph, either alternative should respond to all of your group members’ feedback.

ANSWER

Random Variable

A random variable is a numerical variable that represents the outcomes of a random experiment or process. It assigns a numerical value to each possible outcome, allowing us to quantify and analyze the uncertainty associated with the experiment. It serves as a way to measure and study the probabilistic nature of events.

Discrete Random Variable

A discrete random variable is a type of random variable that can take on a countable or finite number of distinct values. It is characterized by gaps or jumps between values, where only specific outcomes are possible. For example, the number of heads obtained when flipping a coin three times is a discrete random variable, as it can only take on the values of 0, 1, 2, or 3.

Limitations of Probability Prediction

Using a discrete random variable approach to probability prediction has certain limitations. Firstly, it assumes that the outcomes are independent and identically distributed, which may not always be the case in real-world scenarios. Additionally, the approach assumes that the probabilities associated with each outcome remain constant over time, which may not hold true in dynamic or evolving situations. Furthermore, it requires a complete and accurate understanding of the underlying probability distribution, which may be challenging to obtain in practice.

Data Requirement

To use a discrete random variable approach, we need data that represents the outcomes of the random experiment or process. This data should provide information on the possible values that the random variable can take, as well as their associated probabilities. The data can be obtained through empirical observations, historical records, or experimental studies.

Expected Value

The expected value, also known as the mean or average, of a random variable represents the long-term average outcome that we would expect to observe if the random experiment or process is repeated many times. It is calculated by multiplying each possible value of the random variable by its corresponding probability and summing up these products. In simpler terms, the expected value provides a measure of the central tendency or typical outcome of a random variable.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials. It is characterized by two elements: the number of trials (n) and the probability of success in each trial (p). For example, in a clinical trial to test the effectiveness of a new drug, the binomial distribution can be used to model the number of patients who show improvement (success) out of a fixed sample size.

Hypergeometric Distribution

The hypergeometric distribution models the number of successes in a specific sample drawn without replacement from a finite population that contains both successes and failures. It is characterized by three elements: the population size (N), the number of successes in the population (K), and the sample size (n). An example application could be analyzing the number of defective products in a sample drawn from a production line, where the hypergeometric distribution can provide insights into the quality control process.

Negative Binomial Distribution

The negative binomial distribution models the number of independent Bernoulli trials needed to achieve a fixed number of successes. It is characterized by two elements: the number of successes (r) and the probability of success in each trial (p). For instance, in a marketing campaign, the negative binomial distribution can be used to analyze the number of customer interactions required to achieve a specific number of sales.

Relationship with Bernoulli Random Variable

The Bernoulli random variable represents a single trial with two possible outcomes: success (usually denoted as 1) or failure (usually denoted as 0). Both the binomial and negative binomial distributions can be seen as extensions of the Bernoulli distribution. The binomial distribution represents the number of successes in a fixed number of independent Bernoulli trials, while the negative binomial distribution represents the number of trials needed to achieve a fixed number of successes. Thus, the Bernoulli random variable serves as the building block for these distributions, which provide a broader perspective on multiple trials and repeated successes.