Although I cannot allow corrections for Test 3 (originally they were only supposed to be for Test 1!), there will be problems that require recognizing and computing special probabilities in Test 4 and on the Final Exam, and I can use them to possibly improve your scores on Test 4.

In order for this to happen, I need you to do the following:

• Scan or take a CLEAR, LEGIBLE photo of the last two pages of your Test 3. (I need to see the scoring.)

• Post the scan or photo to Piazza as a private note to me.

Note: scans are usually easier to read than photos. There is a pretty good scan app called Genius Scan. I posted instructions for using it on another blog. (You should make sure that the name your save your file under contains your name and Test 3.)

[Latexpage]

NOTE: there will be a quiz on Wednesday, based on one of the problems in Confidence Intervals listed below.

Please see this posts for notes: and also more notes and sources on the CLT here: (including yet another link to the resources explaining how to use your graphing calculator to compute special probabilities): Also make sure that you have done the homework in this post! (as usual)

• Additional practice on the exponential distribution and the Central Limit Theorem:
MAT2572additionalExponentialCLTproblems

The problems to do are marked with either a colored dot or a colored line in the margin, except you should omit #8.65(a) and 8.66(a). I will post the answers shortly. Please do these by next Monday at the latest.

• Problems on Confidence Intervals: in your textbook,

p. 309 # 5.3.1, 5.3.2, 5.3.3, 5.3.4, 5.3.13, 5.3.19

Note: in some of these, data is given and you will have to compute the sample mean from the data: just take the ordinary average of the data. Here is a way to enter the data and get the sample mean on TI calculator.  and here it is for the Casio graphing calculator.

MAT2572section5.3problems

• For practice identifying the various special discrete probability distributions, use these problems.  On first time through, just identify the special distribution (binomial, hypergeometric, poisson, negative binomial, geometric, etc.) and its parameters. Then check your answers against what is given in the last few pages. Testing yourself is one of the best learning methods. You can then go back and test yourself a few days later. Try not to memorize answers based on superficial features of the problem, but rather think about the problem and analyze it to see how it fits with its probability distribution. See these notes:

Math2501DistinguishSpecialProbabilityDistributions

• Please make sure that you have saved your scripts for simulations of the distributions in problems 3.3.1 and 3.3.2, based on the script I did in class and which is also included in this post. You do not have to submit them at this time: we will be using them in a few days to do some hypothesis testing (goodness-of-fit).

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

: answers to the extra practice problems

Math8.1practiceDiscretePartSolns

Please SKIP the following problems, which refer to distributions or theorems we have not discussed in this class: skip #8, 11, 14, 22, 25, 30, 33, 35, 38, 40

On first time through, just identify the special distribution (binomial, hypergeometric, poisson, negative binomial, geometric, etc.) and its parameters. Then check your answers against what is given in the last few pages. Testing yourself is one of the best learning methods. You can then go back and test yourself a few days later. Try not to memorize answers based on superficial features of the problem, but rather think about the problem and analyze it to see how it fits with its probability distribution. See these notes:

Math2501DistinguishSpecialProbabilityDistributions

Math2501-CentralLimitTheorem-slideshow

Math2501-CriticalValuesZ-slideshow

Math2501EstimationConfidenceIntervals

Math2501SamplingDistributionsEstimation

simulations example:

orbs <- c(1:5)

orb_samples <- replicate(10, sample(orbs, 2, replace = TRUE))

X <- apply(orb_samples, 2, max)

info <- hist(X, breaks = c(0.5, 1.5, 2.5, 3.5, 4.5, 5.5))

with(info, text(mids, counts, labels = counts, pos = 1))

#The below makes a relative frequency histogram

h <- hist(X, plot = FALSE)

h$counts <- h$counts/sum(h$counts)

plot(h, freq = TRUE, ylab = “Relative Frequency”)

Datacamp courses: suggestions for snow day makeup

General programming in R:

Intermediate R

Writing Functions in R

Data Visualization in R

Making Reports using R:

Reporting with R Markdown

Probability and Statistics:

Foundations of Probability in R (starting with Chapter 2)

Statistical Modeling in R (part 1)

Financial and Marketing Applications:

Bond Valuation and Analysis in R

Credit Risk Modeling in R

Equity Valuation in R

Financial Trading in R

Introduction to Portfolio Analysis in R

Introduction to R for Finance

Forecasting Product Demand in R

Also please make sure that you have done and checked the homework problems in this post.

• Problems from the textbook: special distributions (continued)

p. 262 #4.4.1, 4.4.3

p. 269 #4.5.1, 4.5.3

p. 274 #4.6.1, 4.6.2, 4.6.6 (challenge!), 4.6.7 (uses the value of gamma of 1/2 from 4.6.6, plus the recursion relation for the gamma function.)

• Problems from the textbook on the CLT:

p. 259 #4.3.33 (I did the first part in class)

Also please read this blog post (and view the video included), which has a very nice explanation of both the normal distribution and the central limit theorem using bunnies and dragons.

• Don’t forget to choose your Datacamp course for the snow day make up 

• Also don’t forget to do the Datacamp assignment on the binomial distribution (pay attention to how the simulation is done: your next programming assignments will be on simulating certain distributions.)

[Latexpage]

Here are notes related to today’s class. Some have been linked previously.

Instructions for using graphing calculators (TI and Casio) to compute special probability distributions

Special probability distributions (summary notes) – was linked previously:

MAT2572RVsDistributions

Distinguishing the special distributions:

Math2501DistinguishSpecialProbabilityDistributions

(This is the version I wrote for my other class: I may or may not have time to rewrite it a bit.)

Explanations of the names of the special distributions

Sampling distributions and the Central Limit Theorem (slideshow): Math2501-CentralLimitTheorem-slideshow

[I have more extensive notes on sampling distributions, statistics, and the three important theorems which I will post after a bit of a rewrite.]

Note: the standard deviation of the sample means is usually referred to as the standard error of the mean, to distinguish it from the standard deviation of the underlying random variable (the population standard deviation)

A statistic is a number that is computed from a random sample. Better: a statistic is a RV whose value depends on a random sample. So far we have two examples: the sample mean and the sample variance (and, by extension, the sample standard deviation is also a statistic). Generally, a statistic is connected to a population parameter which the statistic is intended to estimate: the sample mean $\bar{X}$ is connected to the population mean $\mu$, and the sample variance $S^{2}$ is connected to the population variance $\sigma^{2}$.

[If you are interested in the reason that the sample variance is divided by n-1 rather than n, here is a good discussion from stackexchange. The mathematical reason for it is that this makes the sample variance an “unbiased estimator” of the population variance. A complete mathematical explanation is here.]

Note: “Central” in the Central Limit Theorem (CLT) refers to the mean, which is a measure of “center” of a distribution. So this is a theorem about the limits of distributions of sample means.

We will not dwell on these three theorems in this section for very long. It is important to get the idea of what the CLT says and why it is important; try not to get overwhelmed by the formal notation. A good idea is to try to rephrase it in your own words.

An example I worked out to show what the Central Limit Theorem is saying about the distribution of sample means is in this spreadsheet. This is based on finite populations, but the results are still revealing. Some more examples from populations with very wild distributions are here. There is a demonstration program you can play with here. and some discussion and more examples here.
We will be using simulations in R to illustrate the CLT.

Choose one of the Datacamp courses that you think you might be interested in. A list of suggestions follows below. Each of these courses is estimated to take about 4 hours to complete: you must complete at least half of the course you choose, and take notes while you do that. (Extra credit will be given if you complete the course you have chosen.)

Then you will write a post on the OpenLab site for this class, describing the course that you worked on and something interesting you learned in it. More detail will be given in a later post.

Please choose your course no later than Wednesday 2 May, and make a private post to me in Piazza telling me which course you have chosen. When you are choosing a course, it may help to view the video at the start of the courses (if there is one) for courses you are not sure about.

Then get to work on it, so that you don’t end up doing it all at the last minute!

Datacamp courses: suggestions for snow day makeup

General programming in R:

Intermediate R

Writing Functions in R

Data Visualization in R

Making Reports using R:

Reporting with R Markdown

Probability and Statistics:

Foundations of Probability in R (starting with Chapter 2)

Statistical Modeling in R (part 1)

Financial and Marketing Applications:

Bond Valuation and Analysis in R

Credit Risk Modeling in R

Equity Valuation in R

Financial Trading in R

Introduction to Portfolio Analysis in R

Introduction to R for Finance

Forecasting Product Demand in R

Test 3 is scheduled for the first hour or so of class on Wednesday 25 April.

Review problems are here:

MAT2572Test3ReviewSpring2018

Answers and some hints are here: UPDATE the answer to #3 is now included

MAT2572Test3ReviewAnswersSpring2018

Please let us know on Piazza if you find any typographical or other errors in these!

This test will include the following topics: (My notes on the special distributions are here: MAT2572RVsDistributions)

• Variance and standard deviation of a finite RV

• Expected value (mean), Variance, and standard deviation of a continuous RV

• Binomial distribution

• Hypergeometric distribution

• Poisson distribution

• Estimating binomial probability using Poisson

• Normal distribution

• Estimating binomial probability using normal distribution (including the continuity correction)

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Sampling distribution of the mean

Law of large numbers