Reference no: EM133490706

Question 1 - Hypergeometric Poker

Consider a Poker game where an opponent tells you that the five cards she or he holds (and which you cannot see) represent a hand that beats 60% of all other possible hands. (If you aren't that familiar with Poker, a quick web search will help you understand the relationships between terrible and excellent poker hands -- with better hands having lower probability)

Discuss how you would use that knowledge, along with the Hypergeometric Distribution, to correctly identify the hand your opponent is holding. Be very specific in describing your step by step approach to solving the problem. Exactly what probabilities would you be calculating, in what order, and why? How accurate can you be, and why can't you be more accurate than that?

Question 2 - Binomial & Poisson Fitting

If someone tells us that a random variable is described by a Binomial or Poisson distribution, it becomes a very simple matter (presuming we know the key parameters) to be able to then predict the likelihood that any particular combination of values will be seen in a dataset containing that variable. Sometimes, though, we don't know what kind of distribution our variable will fit, or we don't know enough to be able to identify the key parameters needed to make accurate predictions.

Part of the analysis we routinely do with datasets is to identify whether or not any of the variables included are Binomial or Poisson in nature. Discuss why it can be helpful to do this?

Discuss how you would identify which variables in the dataset would benefit from such analysis. Does it provide benefit to name the distribution that applies even if you can't precisely identify the needed parameters for that distribution?

Use specific examples in your post: Provide some examples of variables that might be in your data that would be considered Binomial or Poisson. These examples should demonstrate that you understand how to recognize that a variable fit one of these distributions.

Question 3 - Predicting Defectives & Defects

Discrete probability distributions are extremely useful in engineering statistics. One particular area where these distributions are commonly used by engineers is quality engineering.

It turns out that if we know the expected defective rate of a process or product, we can make predictions about actual defectives using the Binomial distribution. For example, if we know our defective rate is 15% and we want to produce 100 of something, the Binomial distribution will allow us to predict the probability of different levels of actual defectives that will be seen among those 100. The expect value of 15% tells us that 15 would be the average value over the long run, but it doesn't tell us how many we'll actually see in the next 100 produced. It might be 0, 5, 10, 15, 20, 25, or 30, and the Binomial distribution can predict the probability of each outcome.

It also turns out that if we have a defective process or product, the number of defects in a defective (also known as the defect density) will be predictable using the Poisson distribution. Some defectives will have only 1 defect, while others might have many more. The parameter of interest with the Poisson distribution is the expected arrival rate. If our quality control data tells us that the expected rate of defects in our defective products or processes is 10, then that number is a statement about how many each defective process or product will contain on average in the long run. It'snot a prediction of exactly how many defects the next defective will have. The Poisson distribution gives us the ability to predict the number of defects that we'll see in the next defective, and it will be a value ranging from below to above the expected value.

Discuss the way that probability allows you, as an engineer, to more accurately predict and manage quality in processes and products. Without these distributions, we might know that our defective rate is 15% and that the expected defects in a defective is 10. How would we manage quality if that's all we knew? Alternatively, what do we know about the best case and worst case scenarios if we consider that the Binomial distribution governs the number of defectives we'll see, and the Poisson distribution governs the number of defects we'll see in a defective. How would you use this information to inform management about why sometimes we have good quality days, and other times we have bad quality days? How would you explain that sometimes we have lots of defects but very few defects per defective, while other times we have few defectives but each has lots of defects? How do the Binomial and Poisson distributions help you understand what is happening in each scenario? Use specific examples of probabilities in illustrating your explanations.

Question 4 - Ethics in Engineering (Columbia Space Shuttle Accident)

Background

The Columbia Space Shuttle Accident presents some challenges for engineers. The accident analysis that followed the Space Shuttle Challenger Accident almost 20 years earlier resulted in everyone having a pretty good idea of the reliability of the system. Since Success/Failure of a mission is a binary outcome, the Binomial distribution would govern our understanding of the future outcome so the system based on expected reliability.

Discussion

Discuss the ethics of the circumstances that resulted in the Columbia Space Shuttle Accident. Considering the predictions that were made years before the accident, as well as the reliability of the Binomial distribution and its implications, what could or should the engineers associated with the program have done differently? What obligations do we have as engineers when we find ourselves in this kind of position? Ultimately, why did the system fail, and who shares the responsibility?

Question 5 - Gambler's Fallacy

Describe the Gambler's Fallacy from our readings, and illustrate the concept using an example from the real world where a person or organization might suffer as a result of the fallacy. How would you use an explanation of the fallacy to explain to someone why they were making a mistake in their reasoning? In you respond to others on this topic, comment on whether the explanation they've offered is adequate for addressing the problem.

Unit 4 Topics:

Question 6 - Normal Approximations

This week's and last week's readings have included the Binomial, Poisson and Normal distributions. Each provides very accurate results when used in the circumstances for which they are intended in quality engineering. However, the Binomial and Poisson distributions can get very cumbersome to calculate as the scale of the sample size grows. The tables of values provided in most text books typically don't provide values for very large sample sizes. (The largest sample size supported in our text appendices is 36.) As a result, it is conventional to use the Normal distribution as an approximation of either the Binomial or Poisson distribution as sample sizes grow.

Discuss what it means to use the Normal distribution as an approximation for the Binomial or Poisson distribution. Why does it work? What are the strengths or weaknesses of doing so? (Hint: Think about the center versus the tails.)

Question 7 - Statistics vs. Analysis

Statistical Process Control (SPC) is often described as an applied side of the Binomial, Poisson, and Normal distributions. Many have said that an X-bar control chart is simply a Normal distribution turned on its side. Some argue that our statistics course shouldn't try to cover these applications because they represent an analytical skill rather than a statistical technique.

Discuss, first whether or not you agree with the notion that SPC is an application of what we're learning in this class, and second, whether coverage of the topic is helpful in this class. How might your outcomes after this class be affected - positively or negatively - by whether or not we cover Chapter 16 in the text?

Question 8 - Distribution Parameters

The various distributions in the reading this week and last week entail probability curves that mathematically differ according to a group of parameters. Different parameters allow for specification of the shape, scale, and location of the distribution.

Discuss the role that these parameters play in establishing the characteristic form for each distribution. Consider why our text doesn't discuss the location parameter very much. Why might that be? Also, if we strip away these parameters from the various distribution formulas, what's typically left? Treat this thread as a conceptual exploration, not a detailed mathematical proof.

Since our text uses these parameters but doesn't teach them very much, feel free to use additional outside sources in formulating your response to this thread; but please cite any sources you use.

Question 9 - Why These Distributions?

A classic question of logic is: Which came first, the chicken or the egg? In this class, we might adapt that question for exploring the various probability distributions we've seen this week and last week. Many things we analyze as engineers can be best understood as manifestations in the world of the probabilities these distributions describe. Knowing the appropriate descriptive distribution allows us to make predictions about what has happened, or will happen, in the world.

Discuss how you would explain to a non-engineering manager why, when, and how you would make use of these distributions in order to solve a problem or address an opportunity. What would you need to start, and what you have after your analysis?

Question 10 - Probability Plots

In Unit 3 we learned about discrete probability distributions: Hypergeometric, Binomial, and Poisson. In Unit 4, we added the Normal distribution to our list as the first of many continuous distributions. In this unit, we are adding those other continuous probability distributions. We are seeing that the Exponential, Gamma, Weibull, Lognormal, and Beta distributions are appropriate to certain types of engineering problems. Although our readings have tended to focus the most attention on the Normal distributions, we need to recognize that to solve a particular engineering challenge, any of these distributions might be needed. We need to be able to determine which, if any, of these distributions fits any situation in which we might be doing analysis.

We use a probability plot as a tool to determine if a set of data we are analyzing can reasonably be described by one of our probability distributions. I so, then the standard probabilities associated with the distribution can be used to make predictions about the process or system represented by our data. If not, we have to do some extra math to determine our own probabilities by fitting the data we have to an algebraic function (something we'll do when we get to linear and nonlinear regression) and then integrating that function over our range of interest to determine probabilities. The math isn't that difficult once we know the function, but it is certainly faster and easier to do our work if we can quickly show that one of the distributions we already understand fits our data well enough to use it.

Discuss how a probability plot works, and why we can draw conclusions based on the level of fit we see. If the resulting "fit" isn't perfect (which it very rarely is), what factors do you need to consider in deciding about whether to use a particular distribution to solve your challenge? Describe how you would go about determining the best distribution for a set of data (if there actually is one).