HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Chapter 7 Probability

HAWKES LEARNING SYSTEMS

math courseware specialists

Copyright © 2010 by Hawkes Learning

Systems/Quant Systems, Inc.

All rights reserved.

Chapter 7

Probability Distributions: Information about the Future




Section 7.1 Types of Random Variables

Objectives:

• To define discrete random variables.

• To define continuous random variables.

• To describe probability notation.



• Random variable – a numerical outcome of a random process.

• Probability distribution – a model which describes a specific kind of random process.

• Discrete random variable – a random variable which has a countable number of possible outcomes.

• Continuous random variable – a random variable that can assume any value on a continuous segment(s) of the real number line.

Definitions:





Discrete Random Variables:

• To describe a discrete random variable:

• State the variable.• List all the possible values of the variable.• Determine the probabilities of these values.

Notation for Random Variables:

• Capital letters, such as X, will be used to refer to the random variable, while small letters, such as x, will refer to specific values of the random variable. Often the specific values will be subscripted,

1 2, , ... , .nx x x





Example:

Toss a die and observe the outcome of the toss.

First list the three steps:

• State the variable: X = the outcome of the toss of the die.

• List the possible values: 1, 2, 3, 4, 5, 6. In this case

• Determine the probability of each value.

1 1,x

Value of X Probability161616161616

1

2

3

4

5

62 2,x 3 3,x 4 4,x 5 5,x 6and 6.x





Describing a Continuous Random Variable:

•Time between failure. Calculate the time between installing a brake light in your car and the time the light ceases to work.

• Defining a continuous random variable is very similar to defining a discrete random variable.

• Indentify the random variable:

X = Time between installation and failure.• Indentify the range of values:

Between zero and infinity, note X is measured on a continuous scale.

• Define the probability density: Unknown, but probably would be modeled on historical data and is most likely exponentially distributed.

• Note: for continuous random variables, we specify probabilities with probability density functions.





Chapter Name

Section ## Section Name


Section 7.2 Discrete Probability Distributions

Objectives:

• To describe the characteristics of a discrete random variable.



• Discrete Probability Distribution – all possible values of a random variable with their associated probabilities.

Definition:

Characteristics of Discrete Probability Distributions:

• The sum of all probabilities must equal 1.• The probability of any value must be between 0 and 1, inclusively.





Create a probability distribution for X, the number of heads in four tosses of a coin.

Solution:• To begin, list all possible values of X.• Then, to find the probability distribution, we need to calculate the

probability of each outcome.

Tossing a Coin

x P(X=x) Simple Events

0

1

2

3

4

1

164

166

16

1

16

4

16

TTTT

HTTT

HHHH

TTHT

HHHT HHTH HTHH

TTHH THHT THTH HTHT

THTT

HTTHHHTT

THHH

TTTH

1.0P x



Example:



The probability distribution for the price of a stock thirty days from now is given below. Find the probability the price of the stock with be greater than $56.

Stock Prices

x P(X=x)

54.5 .05

55.0 .10

55.5 .25

56.0 .30

56.5 .20

57.0 .10

1.0P x

Based on the probability distribution, the probability that the stock price will be more than $56 in thirty days is calculated as follows:

56 56.5 57.0P X P X P X

.20 .10

.30



Example:

Solution:




Section 7.3 Expected Value

Objectives:

• To define and describe the expected value of a discrete random variable.



Expected Value:

• The expected value of the random variable X is the mean of the random variable X. It is denoted by E(X).

, where .E X x p x p x P X x





John sells cars. Calculate the expected value of the number of cars John sells per day.

Car Sales

x P(X=x)

0

1

2

3

4

.15

.30

.35

.15

.05

x X xP

Solution:

E X x P X x

0 .15 1 .30

2 .35 3 .15

4 .05

0

0

0.30

0.70

0.45

0.20

0.30 0.70 0.450.20

1.65 1.65E X



Example:



You are trying to decide between two different investment

options. The two plans are summarized in the table

below. The left-hand column for each plan gives the

potential profits, and the right-hand columns give their

respective probabilities. Which plan should you choose?

Example:

Expected Value

Investment A Investment B

Profit Probability Profit Probability

$1200 .1 $1500 .3

$950 .2 $800 .1

$130 .4 –$100 .2

–$575 .1 –$250 .2

–$1400 .2 –$690 .2





Solution:

• It is difficult to determine which plan is better by simply looking at the table. • Let’s use the expected value to compare the plans.

For Investment A:

For Investment B:

E(X) = (1200)(.1) + (950)(.2) + (130)(.4) + (–575)(.1) + (–1400)(.2)

= 120 + 190 + 52 – 57.50 – 280

= $24.50

E(X) = (1500)(.3) + (800)(.1) + (–100)(.2) + (–250)(.2) + (–690)(.2)

= 450 + 80 – 20 – 50 – 138

= $322.00 Best option






Section 7.4 Variance of a Discrete Random Variable

Objectives:

• To define and describe the variance of a discrete random variable.



Variance of a Discrete Random Variable:

-V X x p x 2=

• The standard deviation is computed by taking the square root of the variance:

Standard Deviation= V X

• Variance in investments reflects greater risks.





Variance of a Random Variable

Investment A Investment B

Profit Probability Profit Probability

$1200 .1 $1500 .3

$950 .2 $800 .1

$130 .4 –$100 .2

–$575 .1 –$250 .2

–$1400 .2 –$690 .2

To determine the risk, we need to calculate the variance of each investment.

Determine the Risk:





Solution:For Investment A:


Investment A

Profit Probability

$1200 .1

$950 .2

$130 .4

–$575 .1

–$1400 .2

-x p x 2

24.50A AE X

21200 24.50 0.1

2950 24.50 0.2

2130 24.50 0.4

2575 24.50 0.1

21400 24.50 0.2

138,180.03

171,310.05

4452.10

35,940.03

405,840.05

$755,722.26V X

Standard Deviation 755,722.24 $869.32V X





Solution:

For Investment B:


Investment B

Profit Probability

$1500 .3

$800 .1

–$100 .2

–$250 .2

–$690 .2

-x p x 2

322B BE X

21500 322 0.3

2800 322 0.1

2100 322 0.2

2250 322 0.2

2690 322 0.2

416,305.20

22,848.40

35,616.80

65,436.80

204,828.80

$745,036.00V X

Standard Deviation 745,036.00 $863.15V X





Solution:

Since in terms of risk Investment B is considered the better option because it carries slightly less risk.

755,722.26 745,036.00 ,A BV X V X






Section 7.7 The Binomial Distribution

Objectives:

• To define a Binomial random variable.

• To calculate probabilities using the Binomial distribution.

• To calculate the expected value of a Binomial random variable.

• To calculate the variance of a Binomial random variable.



Definition:

• Binomial experiment – a random experiment which satisfies all of the following conditions.

i) There are only two outcomes on each trial of the experiment. (One of the outcomes is usually referred to as a success, and the other as a failure.)

ii) The experiment consists of n identical trials as described in Condition 1.

iii) The probability of success on any one trial is denoted by p and does not change from trial to trial. (Note that the probability of a failure is 1−p and also does not change from trial to trial.)

iv) The trials are independent.

v) The binomial random variable X is the count of the number of successes in n trials.





Toss a coin 5 times and observe the number of heads. Define the experiment in terms of our definition of a binomial experiment.

i. There are only two outcomes, heads or tails.

ii. The experiment will consist of five tosses of a coin.

(Hence: n = 5.)

iii. The probability of getting a head (success) is and does not

change from trial to trial. (Hence: p = .)

iv. The outcome of one toss will not affect other tosses.v. The variable of interest is the count of the number of heads in

5 tosses.

1

21

2



Example:

Solution:



Toss a coin 4 times and observe the number of heads. Create the probability distribution for the number of heads.

Tossing a Coin

EventsNumber of Heads

Probability

HHTT, HTHT, HTTH, THHT, THTH, TTHH

TTTT

HTTT, THTT, TTHT, TTTH

THHH, HTHH, HHTH, HHHT

HHHH

0

1

2

3

4

1

16

4

16

6

16

4

16

1

16



Example:

Solution:



We will define the binomial probability distribution function as follows:

represents the number of combinations of n objects taken x at a time (without replacement) and is given by

!, where ! 1 2 ...1 and 0!=1.

! !nx

nC n n n n

x n x

1

n xn xxP X x C p p

where the number of trials,n

nxC

the number of successes, andx the probability of success.p



Binomial Probability Distribution Function:

What is the probability of getting exactly 7 tails in 18 coin

tosses?

Example:



Solution:

n = 18, p = .5, x = 7

1

n xn xxP X x C p p

7 18 7

187

1 17

21

2P X C

7 1118! 1 1

11!7! 2 2



A quality control expert at a large factory estimates that 10% of

all batteries produced are defective. If a sample of 20 batteries

are taken, what is the probability that no more than 3 are

defective?

Example:



Solution:

n = 20, p = .1, x = 3, but this time we need to look at the probability that no more than three are defective, which is

P(X ≤ 3).

0 1 33 2P X P X P X P X P X

0 20 1 1920 200 1

2 18 3 1720 202 3

0.1 0.9 0.1 0.9

0.1 0.9 0.1 0.9

C C

C C

0.867





Formulas:

Binomial expected value and variance can be defined with the following formulas.

E X np 1V X np p

Example:

A quality control expert at a large factory estimates that 10% of

all batteries produced are defective. If a sample of 20 batteries

is taken, what is the expected value, variance, and standard

deviation of the number of defective batteries?

Solution:

20, .1n p 20 0 2.1E X

120 0. 11 1 .80.V X

Standard Deviati 1.8on= 1.34V X






Section 7.8 The Poisson Distribution

Objectives:

• To define a Poisson random variable.

• To calculate probabilities using the Poisson distribution.



Definitions:

• Poisson distribution – a discrete probability distribution that uses a fixed interval of time or space in which the number of successes are recorded.

where

• In the Poisson distribution

, for 0,1,2,...!

xeP X x x

x

2.71828..., and

average number of "successes".

e

.E X V X



1. The successes must occur one at a time.

2. Each success must be independent of any other successes.



Poisson Distribution Guidelines:

When calculating the Poisson distribution, round your answers to four decimal places.



Suppose that the dial-up Internet connection at your home goes

out an average of 0.75 times every hour. If you plan to be

connected to the internet for 3 hours one afternoon, what is the

probability that you will stay connected the entire time?

Assume that the dial-up disconnections follow a Poisson

distribution.

Example:



Solution:

x = 0, (0.75)(3) = 2.25

0.1054



A typist averages 1 typographical error per paragraph. If the

document has 4 paragraphs, what is the probability that there

will be less than 5 mistakes?

Example:



Solution:

x < 5, =This time we need to look at the probability that less than five mistakes will occur, which is P(X < 5).

(4)(1) = 4

P(X < 5) = P(X ≤ 4)

0.6288



A fast food restaurant averages 2 incorrect orders every 4 hours.

What is the probability that they will get at least 3 orders wrong in

any given day between 11 AM and 11PM? Assume that fast food

errors follow a Poisson distribution.

Example:



Solution:

x ≥ 3, =

This time we need to look at the probability that at least three wrong orders will occur, which is P(X ≥ 3).

(3)(2) = 6

P(X ≥ 3) = 1 – P(X < 3)

= 1 – P(X ≤ 2)

0.9380



! ! !

0 1 26 6 66 6 6

10 1 2

e e e



Objectives:

• To define a Hypergeometric random variable.

• To calculate probabilities using the Hypergeometric distribution.

• To calculate the expected value of a Hypergeometric random

variable.

• To calculate the variance of a Hypergeometric random variable.


Section 7.9 The Hypergeometric Distribution

• Hypergeometric distribution – a special discrete probability function for problems with a fixed number of dependent trials and a specified number of countable successes.



Definitions:

• When calculating the hypergeometric distribution, round your answers to four decimal places.

, where 0 min ,A N Ax n x

Nn

C CP X x x A n

C

the total number of successes possibleA the size of the population, andN the size of the sample drawn.n



1. Each trial consists of selecting one of the N items in the population and results in either a success or a failure.

2. The experiment consists of n trials.

3. The total number of possible successes in the entire population is A.

4. The trials are dependent. (i.e., selections are made without replacement.)



Hypergeometric Distribution Guidelines:



At the local grocery store there are 50 boxes of cereal on the

shelf, half of which contain a prize. Suppose you buy 4 boxes

of cereal. What is the probability that 3 boxes contain a prize?

Example:



A = 25, x = 3, N = 50, n = 4

25 50 253 4 3

504

3C C

P XC

25 253 1

504

C C

C

2300 25

2303000.2497



Solution:



A produce distributor is carrying 10 boxes of Granny Smith apples

and 8 boxes of Golden Delicious apples. If 6 boxes are randomly

delivered to one local market, what is the probability that at least 4

of the boxes delivered contain Golden Delicious apples?

Example:

Solution:

A = 8, x ≥ 4, N = 18, n = 6

4 5 64P X P X P X P X 8 18 8 8 18 8 8 18 84 6 4 5 6 5 6 6 6

18 18 186 6 6

C C C C C C

C C C

70 45 56 10 28 1

18564 18564 18564

0.2014





Formulas:

Hypergeometric expected value and variance can be defined with the following formulas.

AE X n

N

11

N nA AV X n

N N N





Example:

A produce distributor is carrying 10 boxes of Granny Smith apples and 8 boxes of Golden Delicious apples. If 6 boxes are randomly delivered to one local market, what is the expected value and variance of the distribution?

Solution:

A = 8, N = 18, n = 6

E X8

618

8

3

V X 18 68 8

618 18 18

11

160

153



Documents

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Chapter 7 Probability