# Chapter 4 Problems and Solutions

**Topics:**Random variable, Discrete probability distribution, Probability theory

**Pages:**16 (2910 words)

**Published:**May 5, 2012

12th Edition

Chapter 5 Discrete Probability Distributions

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 5-1

Learning Objectives

In this chapter, you learn: The properties of a probability distribution To compute the expected value and variance of a probability distribution To calculate the covariance and understand its use in finance To compute probabilities from binomial, hypergeometric, and Poisson distributions How to use the binomial, hypergeometric, and Poisson distributions to solve business problems Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 5-2

Definitions Random Variables

A random variable represents a possible numerical value from an uncertain event. Discrete random variables produce outcomes that come from a counting process (e.g. number of classes you are taking). Continuous random variables produce outcomes that come from a measurement (e.g. your annual salary, or your weight). Chap 5-3

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Definitions Random Variables

Random Variables Ch. 5

Discrete Random Variable Continuous Random Variable

Ch. 6

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 5-4

Probability Distributions

Probability Distributions Ch. 5 Discrete Probability Distributions Continuous Probability Distributions Ch. 6

Binomial

Poisson Hypergeometric

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Normal

Uniform Exponential

Chap 5-5

Binomial Probability Distribution

A fixed number of observations, n

e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse

Each observation is categorized as to whether or not the “event of interest” occurred

e.g., head or tail in each toss of a coin; defective or not defective light bulb Since these two categories are mutually exclusive and collectively exhaustive

When the probability of the event of interest is represented as π, then the probability of the event of interest not occurring is 1 - π

Constant probability for the event of interest occurring (π) for each observation

Probability of getting a tail is the same each time we toss the coin Chap 5-6

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Possible Applications for the Binomial Distribution

A manufacturing plant labels items as either defective or acceptable A firm bidding for contracts will either get a contract or not A marketing research firm receives survey responses of “yes I will buy” or “no I will not” New job applicants either accept the offer or reject it Chap 5-7

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

The Binomial Distribution Counting Techniques

Suppose the event of interest is obtaining heads on the toss of a fair coin. You are to toss the coin three times. In how many ways can you get two heads? Possible ways: HHT, HTH, THH, so there are three ways you can getting two heads.

This situation is fairly simple. We need to be able to count the number of ways for more complicated situations.

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 5-8

Counting Techniques Rule of Combinations

The number of combinations of selecting X objects out of n objects is

n! n Cx x! (n x)!

where: n! =(n)(n - 1)(n - 2) . . . (2)(1) X! = (X)(X - 1)(X - 2) . . . (2)(1) 0! = 1 (by definition)

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 5-9

Counting Techniques Rule of Combinations

How many possible 3 scoop combinations could you create at an ice cream parlor if you have 31 flavors to select from? The total choices is n = 31, and we select X = 3.

31! 31! 31 30 29 28! 31 5 29 4,495 31 C3 3!(31 3)! 3!28! 3 2 1 28!...

Please join StudyMode to read the full document