# 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

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

Definitions Random Variables
Random Variables Ch. 5
Discrete Random Variable Continuous Random Variable

Ch. 6

Chap 5-4

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

Binomial
Poisson Hypergeometric

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

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

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.

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)

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!...