Computer Base Module 2 - Reflection

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Business Analytics

Probability: An Introduction to Modeling Uncertainty

Chapter 4

Introduction

Uncertainty is an ever-present fact of life for decision makers.

Much time and effort are spent trying to plan for and respond to uncertainty.

Probability is the numerical measure of the likelihood that an event will occur.

This measure of uncertainty is often communicated through a probability distribution:

Extremely helpful in providing additional information about an event.

Can be used to help a decision maker evaluate possible actions and determine best course of action.

Events and Probabilities

Events and Probabilities (Slide 1 of 6)

A random experiment is a process that generates well-defined outcomes.

By specifying all possible outcomes, we identify the sample space for a random experiment; examples:

A coin toss.

Rolling a die.

An event is defined as a collection of outcomes.

Events and Probabilities (Slide 2 of 6)

Table 4.1: Random Experiments and Experimental Outcomes

Random Experiment	Experimental Outcomes
Toss a coin	Head, tail
Roll a die	1, 2, 3, 4, 5, 6
Conduct a sales call	Purchase, no purchase
Hold a particular share of stock for one year	Price of stock goes up, price of stock goes down, no change in stock price
Reduce price of product	Demand goes up, demand goes down, no change in demand

Events and Probabilities (Slide 3 of 6)

Example: California Power & Light Company (CP&L).

CP&L is starting a project designed to increase the generating capacity of one of its plants in southern California.

Analysis of similar construction projects indicates that the possible completion times for the project are 8, 9, 10, 11, and 12 months.

Events and Probabilities (Slide 4 of 6)

Table 4.2: Completion Times for 40 CP&L Projects

Events and Probabilities (Slide 5 of 6)

The probability of an event is equal to the sum of probabilities of outcomes for the event.

CP&L example: Let C denote the event that the project is completed in 10 months or less, C = {8,9,10}.

The probability of event C, denoted

We can tell CP&L management that there is a 0.70 probability that the project will be completed in 10 months or less.

Some Basic Relationships of Probability

Complement of an Event

Addition Law

Some Basic Relationships of Probability (Slide 1 of 11)

Completion of an Event:

Given an event A, the complement of A is defined to be the event consisting of all outcomes that are not in A.

Figure 5.1 shows what is known as a Venn diagram, which illustrates the concept of a complement:

Rectangular area represents the sample space for the random experiment and contains all possible outcomes.

Circle represents event A and contains only the outcomes that belong to A.

Shaded region of the rectangle contains all outcomes not in event A.

Some Basic Relationships of Probability (Slide 2 of 11)

Figure 4.1: Venn Diagram for Event A

Some Basic Relationships of Probability (Slide 3 of 11)

The probability of an event A can be computed easily if the probability of its complement is known.

Some Basic Relationships of Probability (Slide 4 of 11)

Addition Law:

The addition law is helpful when we are interested in knowing the probability that at least one of two events will occur.

Concepts related to the combination of events:

The union of events.

The intersection of events.

Some Basic Relationships of Probability (Slide 5 of 11)

Given two events A and B, the union of A and B is defined as the event containing all outcomes belonging to A or B or both.

The union of A and B is denoted by

The Venn diagram in Figure 4.2 depicts the union of A and B:

One circle contains all the outcomes of A.

The other circle contains all the outcomes of B.

Some Basic Relationships of Probability (Slide 6 of 11)

Figure 4.2: Venn Diagram for the Union of Events A and B

Some Basic Relationships of Probability (Slide 7 of 11)

The definition of the intersection of A and B is the event containing the outcomes that belong to both A and B.

The intersection of A and B is denoted by

The Venn diagram depicting the intersection of A and B is shown in Figure 4.3:

The area in which the two circles overlap is the intersection.

It contains outcomes that are in both A and B.

Some Basic Relationships of Probability (Slide 8 of 11)

Figure 4.3: Venn Diagram for the Intersection of Events A and B

Some Basic Relationships of Probability (Slide 9 of 11)

The addition law provides a way to compute the probability that event A or event B or both will occur.

Used to compute the probability of the union of two events.

A special case arises for mutually exclusive events:

If the occurrence of one event precludes the occurrence of the other.

If the events have no outcomes in common.

Some Basic Relationships of Probability (Slide 10 of 11)

Figure 4.4: Venn Diagram for Mutually Exclusive Events

Some Basic Relationships of Probability (Slide 11 of 11)

Conditional Probability

Independent Events

Multiplication Law

Bayes’ Theorem

Conditional Probability (Slide 1 of 21)

Conditional probability: When the probability of one event is dependent on whether some related event has already occurred.

Illustration: Lancaster Savings and Loan:

Interested in mortgage default risk.

Interested in whether the probability of a customer defaulting differs by marital status.

Conditional Probability (Slide 2 of 21)

Table 4.3: Subset of Data from 300 Home Mortgages of Customers at Lancaster Savings and Loan

Customer No.	Age	Marital Status	Annual Income	Mortgage Amount	Payments per Year	Total Amount Paid	Default on Mortgage?
1	37	Single	$172,125.70	$473,402.96	24	$581,885.13	Yes
2	31	Single	$108,571.04	$300,468.60	12	$489,320.38	No
3	37	Married	$124,136.41	$330,664.24	24	$493,541.93	Yes
4	24	Married	$79,614.04	$230,222.94	24	$449,682.09	Yes
5	27	Single	$68,087.33	$282,203.53	12	$520,581.82	No
6	30	Married	$59,959.80	$251,242.70	24	$356,711.58	Yes

Conditional Probability (Slide 3 of 21)

Table 4.3: Continued

Customer No.	Age	Marital Status	Annual Income	Mortgage Amount	Payments per Year	Total Amount Paid	Default on Mortgage?
7	41	Single	$99,394.05	$282,737.29	12	$524,053.46	No
8	29	Single	$38,527.35	$238,125.19	12	$468,595.99	No
9	31	Married	$112,078.62	$297,133.24	24	$399,617.40	Yes
10	36	Single	$224,899.71	$622,578.74	12	$1,233,002.14	No
11	31	Married	$27,945.36	$215,440.31	24	$285,900.10	Yes
12	40	Single	$48,929.74	$252,885.10	12	$336,574.63	No
13	39	Married	$82,810.92	$183,045.16	12	$262,537.23	No

Conditional Probability (Slide 4 of 21)

Table 4.3: Continued

Customer No.	Age	Marital Status	Annual Income	Mortgage Amount	Payments per Year	Total Amount Paid	Default on Mortgage?
14	31	Single	$68,216.88	$165,309.34	12	$253,633.17	No
15	40	Single	$59,141.13	$220,176.18	12	$424,749.80	No
16	45	Married	$72,568.89	$233,146.91	12	$356,363.93	No
17	32	Married	$101,140.43	$245,360.02	24	$388,429.41	Yes
18	37	Married	$124,876.53	$320,401.04	4	$360,783.45	Yes
19	32	Married	$133,093.15	$494,395.63	12	$861,874.67	No
20	32	Single	$85,268.67	$159,010.33	12	$308,656.11	No

Conditional Probability (Slide 5 of 21)

Table 4.3: Continued

Customer No.	Age	Marital Status	Annual Income	Mortgage Amount	Payments per Year	Total Amount Paid	Default on Mortgage?
21	37	Single	$92,314.96	$249,547.14	24	$342,339.27	Yes
22	29	Married	$120,876.13	$308,618.37	12	$472,668.98	No
23	24	Single	$86,294.13	$258,321.78	24	$380,347.56	Yes
24	32	Married	$216,748.68	$634,609.61	24	$915,640.13	Yes
25	44	Single	$46,389,75	$194,770.91	12	$385,288.86	No

Conditional Probability (Slide 6 of 21)

Table 4.4: Crosstabulation of Marital Status and if Customer Defaults on Mortgage

Marital Status	No Default	Default	Total
Married	64	79	143
Single	116	41	157
Total	180	120	300

From Table 4.4 or Figure 4.5, the probability that a customer defaults on his or her mortgage is

The probability that a customer does not default on his or her mortgage is

Conditional Probability (Slide 7 of 21)

Figure 4.5: PivotTable for Marital Status and Whether Customer Defaults on Mortgage

Conditional Probability (Slide 8 of 21)

When values give the probability of the intersection of two events, the probabilities are called joint probabilities.

Marginal probabilities are found by summing the joint probabilities in the corresponding row or column of the joint probability table.

Conditional probabilities can be computed as the ratio of joint probability to a marginal probability.

Conditional Probability (Slide 9 of 21)

Table 4.5: Joint Probability Table for Customer Mortgage Prepayments

Conditional Probability (Slide 10 of 21)

Figure 4.6: Using Excel PivotTable to Calculate Conditional Probabilities

Conditional Probability (Slide 11 of 21)

Independent Events:

If the probability of event D is not changed by the existence of event M, then we would say that events D and M are independent events.

Otherwise, the events are dependent.

Conditional Probability (Slide 12 of 21)

Multiplication Law:

Multiplication law can be used to calculate the probability of the intersection of two events.

Based on the definition of conditional probability.

Conditional Probability (Slide 13 of 21)

Special case in which events A and B are independent.

To compute the probability of the intersection of two independent events, simply multiply the probabilities of each event.

Conditional Probability (Slide 14 of 21)

Bayes’ Theorem:

Often begin the analysis with initial or prior probability estimates for specific events of interest.

Then, obtain additional information about events.

Given new information, update the prior probability values by calculating revised probabilities, referred to as posterior probabilities.

Bayes’ theorem provides a means for making these probability calculations.

Conditional Probability (Slide 15 of 21)

Example: A manufacturing firm receives shipments of parts from two different suppliers:

65% of the parts purchased from supplier 1.

35% of the parts purchased from supplier 2.

Quality of purchased parts varies according to their source.

Conditional Probability (Slide 16 of 21)

Table 4.6: Historical Quality Levels for Two Suppliers

Historical data suggest the quality ratings of the two suppliers:

	% Good Parts	% Bad Parts
Supplier 1	98	2
Supplier 2	95	5

Figure 4.7 shows a diagram that depicts the process of the firm receiving a part from one of the suppliers and then discovering that the part is good or bad as a two-step random experiment.

Conditional Probability (Slide 17 of 21)

Figure 4.7: Diagram for Two-Supplier Example

Step 1 shows that the part comes from one of two suppliers and Step 2 shows whether the part is good or bad.

Conditional Probability (Slide 18 of 21)

The process of computing joint probabilities can be depicted in what is called a probability tree.

From left to right through the tree:

The probabilities for each branch at step 1 are prior probabilities.

The probabilities for each branch at step 2 are conditional probabilities.

To find the probability of each experimental outcome, multiply the probabilities on the branches leading to the outcome.

Conditional Probability (Slide 19 of 21)

Figure 4.8: Probability Tree for Two-Supplier Example

Conditional Probability (Slide 20 of 21)

Suppose the parts from the two suppliers are used in the firm’s manufacturing process and a machine breaks while attempting the process using a bad part:

Given the information that the part is bad, what is the probability that it came from supplier 1 and what is the probability that it came from supplier 2?

With the information in the probability tree, Bayes’ theorem can be used to answer these questions.

Conditional Probability (Slide 21 of 21)

Bayes’ theorem is applicable when events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space.

Random Variables

Discrete Random Variables

Continuous Random Variables

Random Variables (Slide 1 of 6)

In probability terms, a random variable is a numerical description of the outcome of a random experiment.

Random variables are quantities whose values are not known with certainty.

A random variable can be classified as being either:

Discrete.

Continuous.

Random Variables (Slide 2 of 6)

Discrete Random Variables:

A random variable that can take on only specified discrete values is referred to as a discrete random variable.

Table 4.7 provides examples of discrete random variables.

Table 4.8 repeats the joint probability table for the Lancaster Savings and Loan data, but with the values labeled as random variables.

Random Variables (Slide 3 of 6)

Table 4.7: Examples of Discrete Random Variables

Random Experiment	Random Variable (x)	Possible Values for the Random Variable
Flip a coin	Face of a coin showing	1 if heads; 0 if tails
Roll a die	Number of dots showing on top of die	1, 2, 3, 4, 5, 6
Contact five customers	Number of customers who place an order	0, 1, 2, 3, 4, 5
Operate a health care clinic for one day	Number of patients who arrive	0, 1, 2, 3, …
Offer a customer the choice of two products	Product chosen by customer	0 if none; 1 if choose product A; 2 if choose product B

Random Variables (Slide 4 of 6)

Table 4.8: Joint Probability Table for Customer Mortgage Prepayments

Random Variables (Slide 5 of 6)

Continuous Random Variables:

A random variable that may assume any numerical value in an interval or collection of intervals is called a continuous random variable.

Technically, relatively few random variables are truly continuous; examples are values related to time, weight, distance, and temperature.

Many discrete random variables have a large number of potential outcomes and so can be effectively modeled as continuous random variables.

Random Variables (Slide 6 of 6)

Table 4.9: Examples of Continuous Random Variables

Discrete Probability Distributions

Custom Discrete Probability Distribution

Expected Value and Variance

Discrete Uniform Probability Distribution

Binomial Probability Distribution

Poisson Probability Distribution

Discrete Probability Distributions (Slide 1 of 21)

The probability distribution for a random variable describes the range and relative likelihood of possible values for a random variable.

For a discrete random variable x, the probability distribution is defined by the probability mass function, denoted by

The probability mass function provides the probability for each value of the random variable.

We can present probability distributions graphically.

Discrete Probability Distributions (Slide 2 of 21)

Figure 4.9: Graphical Representation of the Probability Distribution for Whether a Customer Defaults on a Mortgage

Discrete Probability Distributions (Slide 3 of 21)

Custom Discrete Probability Distribution:

A probability that is generated from observations is called an empirical probability distribution.

An empirical probability is considered a custom discrete probability distribution if it is discrete and the possible values of the random variable have different values:

Useful for describing different possible scenarios that have different probabilities.

Probabilities generated using either the subjective method or the relative frequency method.

Discrete Probability Distributions (Slide 4 of 21)

Table 4.10: Summary Table of Number of Payments Made per Year

Example: The random variable describing the number of mortgage payments made per year by randomly chosen customers.

Discrete Probability Distributions (Slide 5 of 21)

Figure 4.10: Excel PivotTable for Number of Payments Made per Year

Discrete Probability Distributions (Slide 6 of 21)

Expected Value and Variance:

The expected value, or mean, of a random variable is a measure of the central location for the random variable.

Discrete Probability Distributions (Slide 7 of 21)

Table 4.11: Calculation of the Expected Value for Number of Payments Made per Year by a Lancaster Savings and Loan Mortgage Customer

If Lancaster Savings and Loan signs a new mortgage customer, the expected number of payments per year for this customer is 13.8.

Discrete Probability Distributions (Slide 8 of 21)

Figure 4.11: Using Excel SUMPRODUCT Function to Calculate the Expected Value for Number of Payments Made per Year by a Lancaster Savings and Loan Mortgage Customer

Discrete Probability Distributions (Slide 9 of 21)

Figure 4.12: Excel Calculation of the Expected Value for Number of Payments Made per Year by a Lancaster Savings and Loan Mortgage Customer

Discrete Probability Distributions (Slide 10 of 21)

Variance is a measure of variability in the values of a random variable:

Discrete Probability Distributions (Slide 11 of 21)

Table 4.12: Calculation of the Variance for Number of Payments Made per Year by a Lancaster Savings and Loan Mortgage Customer

Discrete Probability Distributions (Slide 12 of 21)

Figure 4.13: Excel Calculation of the Variance for Number of Payments Made per Year by a Lancaster Savings and Loan Mortgage Customer

Discrete Probability Distributions (Slide 13 of 21)

Discrete Uniform Probability Distribution:

When the possible values of the probability mass function are all equal, then the probability distribution is a discrete uniform probability distribution.

Where n = the number of unique values that may be assumed by the random variable.

Discrete Probability Distributions (Slide 14 of 21)

Binomial Probability Distribution:

A binomial probability distribution is a discrete probability distribution that can be used to describe many situations in which a fixed number (n) of repeated identical and independent trials has two, and only two, possible outcomes:

Success.

Failure.

Discrete Probability Distributions (Slide 15 of 21)

The probability mass function for a binomial random variable that calculates the probability of x successes in n independent events.

Discrete Probability Distributions (Slide 16 of 21)

Table 4.13: Probability Distribution for the Number of Customers Who Click on the Link in the Martin’s Targeted E-Mail

Example: Martin’s, an online specialty clothing store, sends out targeted e-mails to its best customers notifying them about special discounts available only to the recipients.

Discrete Probability Distributions (Slide 17 of 21)

Figure 4.14: Graphical Representation of the Probability Distribution for the Number of Customers Who Click on the Link in the Martin’s Targeted E-Mail

Discrete Probability Distributions (Slide 18 of 21)

Figure 4.15: Excel Worksheet for Computing Binomial Probabilities of the Number of Customers Who Make a Purchase at Martin’s

Discrete Probability Distributions (Slide 19 of 21)

Poisson Probability Distribution:

We consider a discrete random variable that is often useful in estimating the number of occurrences of an event over a specified interval of time and space.

Examples:

Number of patients who arrive at a health care clinic in one hour.

Number of computer-server failures in a month.

Number of repairs needed in 10 miles of highway.

Number of leaks in 100 miles of pipeline.

Discrete Probability Distributions (Slide 20 of 21)

If the following two properties are satisfied, the number of occurrences is a random variable described by the Poisson probability distribution:

The probability of an occurrence is the same for any two intervals (of time or space) of equal length.

The occurrence or nonoccurrence in any interval (of time or space) is independent of the occurrence or nonoccurrence in any other interval.

Discrete Probability Distributions (Slide 21 of 21)

Figure 4.16: Excel Worksheet for Computing Poisson Probabilities of the Number of Patients Arriving at the Emergency Room

Continuous Probability Distributions

Uniform Probability Distribution

Triangular Probability Distribution

Normal Probability Distribution

Exponential Probability Distribution

Continuous Probability Distributions (Slide 1 of 30)

Fundamental difference separates discrete and continuous random variables in terms of how probabilities are computed:

Discrete random variables: the probability mass function

provides the probability that the random variable assumes a particular value.

Continuous random variables – the counterpart of the probability mass function is the probability density function, also denoted by

The probability density function does not directly provide probabilities.

We are computing the probability that the random variable assumes any value in an interval.

For continuous random variables, the probability of any particular value of the random variable is zero.

Continuous Probability Distributions (Slide 2 of 30)

Uniform Probability Distribution:

Example: Random variable x representing the flight time of an airplane traveling from Chicago to New York.

With every interval of a given length being equally likely, the random variable x is said to have a uniform probability distribution.

Continuous Probability Distributions (Slide 3 of 30)

Figure 4.17: Uniform Probability Distribution for Flight Time

Continuous Probability Distributions (Slide 4 of 30)

Figure 4.18: The Area Under the Graph Provides the Probability of a Flight Time Between 120 and 130 Minutes

Continuous Probability Distributions (Slide 5 of 30)

The calculation of the expected value and variance for a continuous random variable is analogous to that for a discrete random variable.

For uniform continuous probability distribution, the formulas for the expected value and variance are:

Continuous Probability Distributions (Slide 6 of 30)

Triangular Probability Distribution:

Useful only when subjective probability estimates are available.

In the triangular probability distribution, we need only specify:

The minimum possible value a.

The maximum possible value b.

The most likely value (or mode) of the distribution m.

If these values can be knowledgeably estimated, then as an approximation of the actual probability density function, we can assume that the triangular distribution applies.

Continuous Probability Distributions (Slide 7 of 30)

Figure 4.19: Triangular Probability Distribution for Time Required for Initial Assessment of Corporate Headquarters Construction

Continuous Probability Distributions (Slide 8 of 30)

Note in Figure 4.19 that the probability density function is a triangular shape.

The general form of the triangular probability density function is:

Continuous Probability Distributions (Slide 9 of 30)

The geometry required to find the area under the graph for any given value is slightly more complex than that required to find the area for a uniform distribution.

Continuous Probability Distributions (Slide 10 of 30)

Figure 4.20: Triangular Distribution to Determine

Continuous Probability Distributions (Slide 11 of 30)

Normal Probability Distribution:

One of the most useful probability distributions for describing a continuous random variable is the normal probability distribution.

Wide variety of practical and business applications:

Heights and weights of people.

Test scores.

Scientific measurements.

Uncertain quantities such as demand for products.

Rate of return for stocks and bonds.

Time it takes to manufacture a part or complete an activity.

Continuous Probability Distributions (Slide 12 of 30)

Figure 4.21: Bell-Shaped Curve for the Normal Distribution

Continuous Probability Distributions (Slide 13 of 30)

The probability density function that defines the bell-shaped curve of the normal distribution is:

Continuous Probability Distributions (Slide 14 of 30)

Characteristics of the normal distribution:

The entire family of normal distributions is differentiated by two

The highest point on the normal curve is at the mean, which is also the median and mode of the distribution.

The mean of the distribution can be any numerical value: negative, zero, or positive (see Figure 4.22).

Continuous Probability Distributions (Slide 15 of 30)

Figure 4.22: Three Normal Distributions with the Same Standard Deviation but Different Means

Continuous Probability Distributions (Slide 16 of 30)

Characteristics of the normal distribution (continued):

The normal distribution is symmetric, with the shape of the normal curve to the left of the mean a mirror image of the shape of the normal curve to the right of the mean.

The tails of the curve extend to infinity in both directions and theoretically never touch the horizontal axis.

The standard deviation determines how flat and wide the normal curve is; larger values of the standard deviation result in wider, flatter curves, showing more variability in the data (see Figure 4.23).

Continuous Probability Distributions (Slide 17 of 30)

Figure 4.23: Two Normal Distributions with the Same Mean but Different Standard Deviations

Continuous Probability Distributions (Slide 18 of 30)

Characteristics of the normal distribution (continued):

Probabilities for the normal random variable are given by areas under the normal curve. The total area under the curve for the normal distribution is 1. Because the distribution is symmetric, the area under the curve to left of the mean is 0.50 and the area to the right is 0.50.

The percentages of values in some commonly used intervals are:

68.3% of the values of a normal random variable are within plus or minus one standard deviation of its mean.

95.4% of the values of a normal random variable are within plus or minus two standard deviations of its mean.

99.7% of the values of a normal random variable are within plus or minus three standard deviations of its mean.

Continuous Probability Distributions (Slide 19 of 30)

Figure 4.24: Areas Under the Curve for Any Normal Distribution

Continuous Probability Distributions (Slide 20 of 30)

Application of the normal probability distribution: Grear Aircraft Engines sells aircraft engines to commercial airlines.

Grear offers performance-based sales contract guaranteeing that engines will provide certain amount of lifetime flight hours subject to airline purchasing a preventive-maintenance service plan.

Based on extensive flight testing and computer simulations, Grear believes mean lifetime flight hours is normally distributed with a mean

What is the probability that an engine will last more than 40,000 hours?

Continuous Probability Distributions (Slide 21 of 30)

Figure 4.25: Grear Aircraft Engines Lifetime Flight Hours Distribution

Continuous Probability Distributions (Slide 22 of 30)

Figure 4.26: Excel Calculations for Grear Aircraft Engines Example

Continuous Probability Distributions (Slide 23 of 30)

Grear is considering a guarantee that will provide a discount on a replacement aircraft engine if the original engine does not meet the lifetime-flight-hour guarantee.

How many lifetime flight hours should Grear guarantee if Grear wants no more than 10% of aircraft engines to be eligible for the discount guarantee? (See Figure 4.27.)

How do we calculate the probability that an engine will have a lifetime of flight hours greater than 30,000 but less than 40,000 hours? (See Figures 4.28 and 4.29.)

Continuous Probability Distributions (Slide 24 of 30)

Figure 4.27: Grear’s Discount Guarantee

Continuous Probability Distributions (Slide 25 of 30)

Figure 4.28: Graph Showing the Area Under the Curve Corresponding to

in the Grear Aircraft Engines Example

Continuous Probability Distributions (Slide 26 of 30)

Figure 4.29: Using Excel to Find

Aircraft Engines Example

Continuous Probability Distributions (Slide 27 of 30)

Exponential Probability Distribution:

The exponential probability distribution may be used for random variables such as:

Time between patient arrivals at an emergency room.

Distance between major defects in a highway.

Time until default in certain credit-risk models.

100

Continuous Probability Distributions (Slide 28 of 30)

Example: x represents time between business loan defaults for a particular lending agency.

If the mean time between loan defaults is 15 months, the graph of the probability is shown in Figure 4.30.

To compute exponential probabilities, we use:

Figure 4.31 shows how to calculate these values for an exponential distribution in Excel using EXPON.DIST.

101

Continuous Probability Distributions (Slide 29 of 30)

Figure 4.30: Exponential Distribution for the Time Between Business Loan Defaults Example

102

Continuous Probability Distributions (Slide 30 of 30)

Figure 4.31: Using Excel to Calculate

Business Loan Defaults Example

103

End of Chapter 4

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