IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS

1. IAM 530ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION

2. WHAT IS STATISTICS? • Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics is a way to get information from data. It is the science of uncertainty.

3. WHAT IS STATISTICS? • A pharmaceutical CEO wants to know if a new drug is superior to already existing drugs, or possible side effects. • How fuel efficient a certain car model is? • Is there any relationship between your GPA and employment opportunities? • Actuaries want to determine “risky” customers for insurance companies.

4. STEPS OF STATISTICAL PRACTICE • Preparation: Set clearly defined goals, questions of interests for the investigation • Data collection: Make a plan of which data to collect and how to collect it • Data analysis: Apply appropriate statistical methods to extract information from the data • Data interpretation: Interpret the information and draw conclusions

5. STATISTICAL METHODS • Descriptive statistics include the collection, presentation and description of numerical data. • Inferential statistics include making inference, decisions by the appropriate statistical methods by using the collected data. • Model building includes developing prediction equations to understand a complex system.

6. BASIC DEFINITIONS • POPULATION: The collection of all items of interest in a particular study. • SAMPLE: A set of data drawn from the population; • a subset of the population available for observation • PARAMETER: A descriptive measure of the • population, e.g., mean • STATISTIC: A descriptive measure of a sample • VARIABLE: A characteristic of interest about each • element of a population or sample.

7. EXAMPLE PopulationUnitSample Variable All students currently Student Any departmentGPA enrolled in schoolHours of works per week All books in library BookStatistics’ BooksReplacement cost Frequency of check out Repair needs All campus fast food RestaurantBurger King Number of employees restaurants Seating capacity Hiring/Not hiring Note that some samples are not representative of population and shouldn’t be used to draw conclusions about population.

8. How not to run a presidential poll For the 1936 election, the Literary Digest picked names at random out of telephone books in some cities and sent these people some ballots, attempting to predict the election results, Roosevelt versus Landon, by the returns. Now, even if 100% returned the ballots, even if all told how they really felt, even if all would vote, even if none would change their minds by election day, still this method could be (and was) in trouble: They estimated a conditional probability in that part of the American population which had phones and showed that that part was not typical of the total population. [Dudewicz & Mishra, 1988]

9. STATISTIC • Statistic (or estimator) is any function of a r.v. of r.s. which do not contain any unknown quantity. E.g. • are statistics. • are NOT. • Any observed or particular value of an estimator is an estimate.

10. RANDOM VARIABLES • Variables whose observed value is determined by chance • A r.v. is a function defined on the sample space S that associates a real number with each outcome in S. • Rvs are denoted by uppercase letters, and their observed values by lowercase letters. • Example: Consider the random variable X, the number of brown-eyed children born to a couple heterozygous for eye color (each with genes for both brown and blue eyes). If the couple is assumed to have 2 children, X can assume any of the values 0,1, or 2. The variable is random in that brown eyes depend on the chance inheritance of a dominant gene at conception. If for a particular couple there are two brown-eyed children, we have x=2.

11. COLLECTING DATA • Target Population: The population about which we want to draw inferences. • Sampled Population: The actual population from which the sample has been taken.

12. SAMPLING PLAN • Simple Random Sample (SRS): All possible samples with the same number of observations are equally likely to be selected. • Stratified Sampling: Population is separated into mutually exclusive sets (strata) and then sample is drawn using simple random samples from each strata. • Convenience Sample: It is obtained by selecting individuals or objects without systematic randomization.

14. EXAMPLE • A manufacturer of computer chips claims that less than 10% of his products are defective. When 1000 chips were drawn from a large production run, 7.5% were found to be defective. • What is the population of interest? • What is the sample? • What is parameter? • What is statistic? • Does the value 10% refer to a parameter or a statistics? • Explain briefly how the statistic can be used to make inferences about the parameter to test the claim. The complete production run for the computer chips 1000 chips Proportion of the all chips that are defective Proportion of sample chips that are defective Parameter Because the sample proportion is less than 10%, we can conclude that the claim may be true.

15. DESCRIPTIVE STATISTICS • Descriptive statistics involves the arrangement, summary, and presentation of data, to enable meaningful interpretation, and to support decision making. • Descriptive statistics methods make use of • graphical techniques • numerical descriptive measures. • The methods presented apply both to • the entire population • the sample

16. Types of data and information • A variable - a characteristic of population or sample that is of interest for us. • Cereal choice • Expenditure • The waiting time for medical services • Data - the observed values of variables • Interval and ratio data are numerical observations (in ratio data, the ratio of two observations is meaningful and the value of 0 has a clear “no” interpretation. E.g. of ratio data: weight; e.g. of interval data: temp.) • Nominal data are categorical observations • Ordinal data are ordered categorical observations

17. Types of data – examples

18. Types of data – analysis • Knowing the type of data is necessary to properly select the technique to be used when analyzing data. • Types of descriptive analysis allowed for each type of data • Numerical data – arithmetic calculations • Nominal data – counting the number of observation in each category • Ordinal data - computations based on an ordering process

19. Types of data - examples Numerical data Nominal Age - income 55 75000 42 68000 . . . . PersonMarital status 1 married 2 single 3 single . . . . Weight gain +10 +5 . . Computer Brand 1 IBM 2 Dell 3 IBM . . . .

20. Types of data - examples Numerical data Nominal data A descriptive statistic for nominal data is the proportion of data that falls into each category. Age - income 55 75000 42 68000 . . . . Weight gain +10 +5 . . IBM Dell Compaq Other Total 25 11 8 6 50 50% 22% 16% 12%

21. Cross-Sectional/Time-Series/Panel Data • Cross sectional data is collected at a certain point in time • Test score in a statistics course • Starting salaries of an MBA program graduates • Time series data is collected over successive points in time • Weekly closing price of gold • Amount of crude oil imported monthly • Panel data is collected over successive points in time as well

22. Differences

23. GAMES OF CHANCE

24. COUNTING TECHNIQUES • Methods to determine how many subsets can be obtained from a set of objects are called counting techniques. FUNDAMENTAL THEOREM OF COUNTING If a job consists of k separate tasks, the i-th of which can be done in ni ways, i=1,2,…,k, then the entire job can be done in n1xn2x…xnk ways.

25. THE FACTORIAL • number of ways in which objects can be permuted. n! = n(n-1)(n-2)…2.1 0! = 1, 1! = 1 Example: Possible permutations of {1,2,3} are {1,2,3}, {1,3,2}, {3,1,2}, {2,1,3}, {2,3,1}, {3,2,1}. So, there are 3!=6 different permutations.

26. COUNTING • Partition Rule: There exists a single set of N distinctly different elements which is partitioned into k sets; the first set containing n1 elements, …, the k-th set containing nk elements. The number of different partitions is

27. COUNTING • Example: Let’s partition {1,2,3} into two sets; first with 1 element, second with 2 elements. • Solution: Partition 1: {1} {2,3} Partition 2: {2} {1,3} Partition 3: {3} {1,2} 3!/(1! 2!)=3 different partitions

28. Example • How many different arrangements can be made of the letters “ISI”? 1st letter 2nd letter 3rd letter I I S S I S I I N=3, n1=2, n2=1; 3!/(2!1!)=3

29. Example • How many different arrangements can be made of the letters “statistics”? • N=10, n1=3 s, n2=3 t, n3=1 a, n4=2 i, n5=1 c

30. COUNTING (e.g. picking the first 3 winners of a competition) • Ordered, without replacement • Ordered, with replacement 3. Unordered, without replacement 4. Unordered, with replacement (e.g. tossing a coin and observing a Head in the k th toss) (e.g. 6/49 lottery) (e.g. picking up red balls from an urn that has both red and green balls & putting them back)

31. PERMUTATIONS • Any ordered sequence of r objects taken from a set of n distinct objects is called a permutation of size r of the objects.

32. COMBINATION • Given a set of n distinct objects, any unordered subset of size r of the objects is called a combination. Properties

33. COUNTING

34. EXAMPLE • How many different ways can we arrange 3 books (A, B and C) in a shelf? • Order is important; without replacement • n=3, r=3; n!/(n-r)!=3!/0!=6, or

35. EXAMPLE, cont. • How many different ways can we arrange 3 books (A, B and C) in a shelf? 1st book2nd book3rd book A B C C B B A C C A B C A B A

36. EXAMPLE • Lotto games: Suppose that you pick 6 numbers out of 49 • What is the number of possible choices • If the order does not matter and no repetition is allowed? • If the order matters and no repetition is allowed?