Lecture 1 (Part I) Material Covered in This Lecture: Chapter 1, Chapter 2 (2.1 --- 2.6). Chapter 1 1. What is Statistics? 2. Two definitions. (1). Population (2). Sample 3. The objective of statistics. 4. How to represent the data? (1). Classification of data a. Categorical Data b. Numerical Data (2) Methods to represent the data a. Graphical Method
b. Numerical Method Measures of central tendency Measures of variation c. Empirical Rule
Chapter 2 1. Some descriptive definition of Probability : Probability of an event tells us how likely the event will happen; Probability is a measure of one s belief in the occurrence of a future event. Examples: Tossing a balanced coin; Rolling a fair die; Tomorrow s temperature, etc. 2. Basic set notations and theory What is set? A collection of elements. Some particular sets. Universal Set S: Empty Set f: Relationships between sets and Venn diagrams. Relationship Venn diagram Subset (A Õ B) Union (A» B) Intersection (A B) Complement ( A ) Disjoint (A B = f)
Four equalities of sets (1). Distributive Laws. A (B» C ) = (A B)» (A C) A» (B C ) = (A» B) (A» C) (2). DeMorgan s Laws. ( AI B) = AU B ( AU B) = AI B 3. Some basic definition of Probability (1). Experiment: An experiment is the process by which an observation is made. (2). Sample Space: The collection of all outcomes associated with an experiment. (Notation: S) Discrete Sample Space: A discrete sample space is one that contains either a finite or a countable number of distinct sample points. Non-Discrete Sample Space will be discussed in Chapter 4. (3). Event: Any subset of the sample space is called an event. (Notation: A, B, C, ) (4). Simple Event or Sample Point: An event that cannot be decomposed. (Notation: E 1, E 2, ) Distinct simple events are disjoint. (5). Compound Event: An event which can be decomposed into other events. (6). Occurrence of an Event: An event occurs if and only if one of the sample points in A occurs.
4. Probability. Suppose S is a sample space associated with an experiment. To every event A in S we assign a number, P(A), called the probability of A, so that the following axioms hold: Axiom 1: P(A) 0. Axiom 2: P(S) = 1. Axiom 3: If A 1, A 2, A 3, form a sequence of pairwise disjoint events in S (that is, A i A j = f if i j), then P( A Example 1: Tossing a balanced coin. 1 U A2 U A3 A i ) = U...) P( i=1 Example 2: Rolling a fair die. Example 3 (Example 2.1). A manufacturer has five seemingly identical computer terminals available for shipping. Unknown to her, two of the five are defective. A particular order calls for two of the terminals and is filled by randomly selecting two of the five that are available. a. List the sample space for this experiment. b. Let A denote the event that the order is filled with two nondefective terminals. List the sample points in A. c. Construct a Venn diagram for the experiment that illustrates event A. d. Assign probabilities to the simple events in such a way that the information about the experiment is used and the axioms in Definition 2.6 are met.
e. Find the probability of A. 5. Methods of calculating the probability (Discrete Sample Case) (1). The Sample Point Method. The following steps are used to find the probability of an event. a. Simple Event: Define the experiment and clearly determine how to describe one simple event.. b. Sample Space: List the simple events associated with the experiment and test each to make certain that it cannot be decomposed. This defines the sample space S. c. Probabilities of Simple Events: Assign reasonable probabilities to the sample points in S, making certain that P(E i ) 0 and SP(E i ) = 1. d. Event of interest: Define the event of interest, A, as a specific collection of sample points. e. Probability of Event of interest: Find P(A) by summing the probabilities of the sample points in A. Example 2.2: Consider the problem of selecting two applicants for a job out of a group of five and imagine that the applicants vary in competence, 1 being the best, 2 second best, and so on, for 3,4 and 5. These ratings are, of course, unknown to the employer. Define two events A and B as A: The employer selects the best and one of the two poorest applicants. B: The employer selects at least one of the two best. Find the probabilities of these events.
Example 2.4: The odds are two to one that, when A and B play tennis, A wins. Suppose that A and B play two matches. What is the probability that A wins at least one match. Tools for counting sample points. When the number of simple events in a sample space is very large and manual enumeration of every sample point is tedious or even impossible, counting the number of points in the sample space and in the event of interest may be the only efficient way to calculate the probability of an event. This section presents some useful results from the theory of combinatorial analysis. mn Rule: With m elements a 1, a 2,, a m and n elements b 1, b 2,, b n it is possible to form mn=m ä n pairs containing one element from each group. Example 2.5: An experiment involves tossing a pair of dice and observing the numbers on the upper faces. Find the number of sample points in S. Generalization of mn Rule: Example 3.6: Tossing a coin three times and observing the upper faces. Find the number of sample points in S.
Example 3.7: Consider an experiment that consists of recording the birthday for each of 20 randomly selected persons. Ignoring leap years and assuming that there are only 365 possible distinct birthdays, find the number of points in the sample space S for this experiment. If we assume that each of the possible sets of birthdays in equiprobable, what is the probability that each person in the 20 has a different birthday? Permutation Rule: An ordered arrangement of r distinct objects is called a permutation. The number of ways of ordering n distinct objects taken r at a time will be designated by the symbol n P r, and P n r = n( n 1)( n 2)...( n r + 1) = n!. ( n r )! Example 2.8: The names of 3 employees are to be randomly drawn, without replacement, from a bowl containing the names of 30 employees of a small company. The person whose name is drawn first receives $100, and the individuals whose names are drawn second and third receive $50 and $25, respectively. How many sample points are associated with this experiment?