Exam 1 Study Guide

General

The exam will be open book, open notes, and open computer. You may use any resources except, of course, asking another person for help (in person or online). You do not need to memorize any formulas.

The emphasis will be on practical applications. There will not be any in-depth theory questions, but you should know how to apply major theoretical concepts such as how to compute the probability of the union of two non-disjoint events (Example 2.14) or conditional probabilities (Example 2.25).

Chapter 2

2.1 Sample Spaces and Events

You should be familiar with the following definitions:

• sample space
• event
• simple event
• compound event
• set union
• set compliment
• set intersection
• mutually exclusive or disjoint events

For example, the probability of rolling 12 with a pair of balanced dice is 1/36 or 0.0278. If we roll the dice a million times, we expect the proportion of twelves in those 1,000,000 outcomes to be close to 0.0278, that is, we expect about 27,778 twelves.

2.2 Probability (abstract definitions)

You should be familiar with the following definitions, axioms, and theorems:

• Axiom 1: The probability of any event is greater than or equal to zero
• Axiom 2: The probability of the entire sample space is one
• Axiom 3: The probability of the union of a collection of disjoint events (i.e., the probability that one or more of the events occur) is the sum of the probabilities of the individual events in the collection.
• Propositions:
  • The probability of the null event (the event containing no outcomes) is zero
  • The probability of any event and the probability of its compliment add up to one
  • The probability of any event is less than or equal to 1
• The probability of the union of two or three non-disjoint events (p.55,56)
• The probability of events in experiments with equally likely outcomes (p.57)

You should be able to use the properties of probability relating to compliments and unions on pages 54 and 55.

2.3 Counting Tecniques

You should be familiar with the following definitions:

• The product rule(s)
• permutations
• combination

You should be able to use a spreadsheet to compute permutations and combinations, understand the difference between them and be able to determine which one to use in a specific problem

You should be able to apply the product rules for ordered pairs and k-tuples (Examples 2.17 and 2.19)

2.4 Conditional Probability

You should be able to apply the definition of conditional probability (Example 2.25), the law of total probability, and Baye's Theorem (Example 2.30).

You should be able to construct a tree diagram and use it to solve a problem (Similar to Examples 2.29 and 2.30 in the text and the ones in written assignment 1)

2.5 Independence

• You should be able to apply the definition of independence and the multiplication rule(s) for intersections (Example 2.23) for two or more events.
• You should be able to construct a tree diagram similar to example 2.34

Chapter 1

Measures of Location

You should be familiar with the following definitions, and able to use a spreadsheet to compute them for a given set of data values (the examples assume the relevant data values are in cells A1:A10 of a spreadsheet, or in an array called x in R)

You should know which measures are sensitive to outliers.

Measures of Variability

You should be familiar with the following definitions, and able to use a spreadsheet to compute them for a given set of data values:

You should know which measures are sensitive to outliers.

Chapter 3

3.1 Random Variables

You should understand the abstract definition of a random variable as a rule or function that associates a number with each outcome in a sample space.

You should be familiar with the definition of a Bernoulli random variable.

3.2 Discrete Random Variables

You should understand the meaning of following terms:

• probability distribution (or probability mass function)
• parameter of a probability distribution
• cumulative distribution function

3.3 Expected values

You should understand the meaning of following terms:

• expected value (or mean value) of a discrete random variable
• variance of a discrete random variable
• standard deviation of a discrete random variable

3.4 The Binomial distribution

You should:

• Understand the conditions of the "binomial experiment"
• Understand the informal definition of a binomial random variable as "the number of successes in n trials"
• Know that the sample space for the binomial experiment is the set {0,1,2,...,n}
• Know the meaning of the parameters n and p
• Know how to determine the mean and variance given n and p

3.5 The Geometric distribution

You should:

• Understand that the geometric distribution results from conducting independent Bernoulli trials until the th "success"
• Know that the sample space for the geometric experiment is the set {0,1,2,3,...}
• Know the meaning of the parameter p
• Know how to determine the mean and variance given the parameter p

3.5 The Negative Binomial distribution

You should:

• Understand that the negative binomial distribution results from conducting independent Bernoulli trials until the r^th "success"
• Understand that the negative binomial distribution can be thought of as the sum of r independent geometric random variables.
• Know that the sample space for the negative binomial experiment is the set {0,1,2,3,...}
• Know the meaning of the parameters r and p
• Know how to determine the mean and variance given r and p

3.6 The Poisson distribution

You should:

• Understand that the Poisson arises as a limit of the binomial distribution as n becomes large and p becomes small while np remains constant
• Understand that very often independent arrival phenomena such as calls arriving to a telephone exchange or cars arriving at a toll plaza can be modeled well by a Poisson distribution
• Know that the sample space for the Poisson experiment is the set {0,1,2,3,...}
• Know the meaning of the parameter lambda
• Know how to determine the mean and variance given lambda