NA387 W 07 Course Summary

Review for Final Exam

Closed book, 2 (2-sided) Sheets allowed

What did we learn ? Got an understanding of probabilistic

aspects of the world of engineering. Learned to analyze various types of

data and problems Most of material was foundation

training, basics and fundamentals. Other courses (IOE, other Depts) cover applications and more in-depth theories.

My Advice is to save everything-

Especially Textbooks, lecture Notes, PPTS, formula sheets, even assignments and exams.

You will most likely need to refer to them again and again when you take higher level courses, but also in many other courses requiring Probs and Stats.

What we learned:

Understanding Events, Probabilities, Distributions, Density Functions.

Selecting the most appropriate distributions for analytical modeling

Selecting the best parameter estimates for Statistics

Understanding the effects of sample size and sampling errors

Following is a more detailed, Chapter by Chapter list.

Chapter 1: Descriptive Statistics

Understand and apply descriptive statistics (mean, standard deviation, variance, range, median)

Understand and apply basic graphical techniques (histogram, dot plot, frequency table)

Chapter 2: Basic Probability

concepts

Understand and apply the basic concepts of Probability Theory (Events, probabilities, intersections, unions, conditional probability, independence of events, Bayes theorem, permutations and combinations)

Chapter 3: Discrete RVs

Discrete Random Variables, PMFs Expected Values, Variances, Conditional EVs and Variances! Bernoulli and Binomial PMFs Geometric, Hypergeometric,

and negative Binomial PMFs Poisson PMF!

Chapter 4: Continuous RVs

Continuous Random Variables Probability Distributions (pdf, cdf)

Uniform Distribution Percentiles

Expected Values and Variance Exponential PDF and Poisson PMF!

Name Description Parameters f(x) F(x) E(X)/V(X)

Uniformconstant (flat) probability in

the interval A and BA, B

NormalMost important distribution. Symmetric about the mean

(normal curve)

GammaVariety of skewed

distributions

Exponential

Special case of Gamma and Weibull. Typically models

time between events. Constant failure rate

Chi-Squared

Basis for some statistical inference procedures

(related to the sigmas of r.v that follow normal dist)

Weibull

Another distribution with wide variety of shapes, can

replicate normal and exponential

LognormalTransformed variable

(Y=Ln(X)) follows a Normal distribution

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Chapter 4 (cont’d)

Normal Distribution!!! Properties, pdf, cdf Standardizing a variable (Must be an

expert with the table!) Percentiles, probabilities… Transform back to original units

Normal Approximations Binomial

Weibull Distribution Pdf, cdf, E(X), V(X), MTTF Exponential also a special case of Weibull

Lognormal Distribution Pdf, cdf, E(X), V(X) Transformation back to original units

Probability Plots Beta Distribution

Pdf, cdf, E(X), V(X)

Chapter 4 -end

Chapter 5: Joint Distributions, Central Limit

Theorem

Jointly distributed variables Discrete Continuous Mixture experiments

Joint Distributions (2 independent random variables)

Expected values; Conditional Expectations

Covariance and Correlation

Chapter 5 (con’d)

Statistics and their distributions Point Estimate – sampling distribution Independent and identically

distributed (iid) random samples Deriving sampling distribution of a

statistic By probability Simulation

Chapter 5 (end) Distribution of the sample mean Central Limit Theorem! Distribution of a linear combination as we stressed in the lectures, the

CLT is still valid if the RVs are independent, even if they are NOT identically distributed, or with same means or variances.

Chapter 6: Point Estimation

Point Estimation Concepts, estimator bias and variance MVUE (minimum variance unbiased

estimator) Standard Error Method of Moments Maximum likelihood Estimation (MLE)

Chapter 7: Confidence Intervals

Given a statistic, generate a confidence interval

mean, proportion, varianceLarge sample CI’s for a Population Mean

and ProportionCI’s based on a Normal PopulationCI’s for Variance and St. Dev of a Normal

PopulationUnderstand effects of sample size

Chapter 8: Single Sample Hypothesis

Testing

Understand effects of sample size and sampling error (type I and II errors) and their relative importance, on one-sample statistical decisions.

Know how to properly conduct a single sample hypothesis test

Tests about a Population Mean, Proportions.

P-values

Chapter 12: Simple Linear Regression and

Correlation Did only Sections 12.1 and 12.5 in detail

12.1: The Simple Linear Regression Model

12.5: Correlation

Chapter 14: Goodness-of-fit Tests

Briefly discussed the Chi-square goodness of fit test, comparing Histograms of data and “Theory” (H0)

Discussed the K-S goodness of for test, comparing cumulatives of data and ‘theory’ (not in text)

Introduced “Delphi” surveys (not in text)

Final Exam: The important things

To prepare for the exam, study:

1. Lectures PPTs, and esp. examples done on the board in lectures, and in the labs

2. Textbook chapters 1-8, Hwks 1-8. , and elements of 12 and 14 (see previous slide)

Questions will be a mix of multi choice-fill in the blanks-etc short answer problems, and full solution simple problems

Final Exam-Con’d

Closed book, alternate seating, two classrooms

Problems will be mostly similar to previous exams, homeworks, etc.

Prepare Well / Good Luck!

In Conclusion:

NA387: An important, basic course that is necessary in many future courses.

Lots of new and very useful knowledge.

Let us know (e-mail?) when you have an opportunity to use Probs and Stats in the future!