Page 1

Title Slide

• Discrete Probability Distributions

• Chapter 5

Page 2

Learning Objectives

In this chapter, you will learn:

• Properties of a probability distribution

• How to calculate expected value and variance of a probability distribution

• How to calculate covariance and understand its use in finance

• How to calculate probabilities for the following distributions: Binomial, Hypergeometric, and Poisson

• How to use Binomial, Hypergeometric, and Poisson distributions to solve business problems

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Definitions

• Discrete Variables: Outcomes arise from a counting process (e.g., the number of courses you choose to take).

• Continuous Variables: Results arise from a measurement (e.g., your annual salary or your weight).

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Types of Variables

• Discrete Variables

• Continuous Variables

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Discrete Variables

• Can take a measurable number of values.

• Examples:

  • Rolling a die twice: Let X be the number of times 4 occurs (X could be 0, 1, or 2).

  • Flipping a coin 5 times: Let X be the number of "heads" (X = 0, 1, 2, 3, 4, or 5).

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Probability Distribution of a Discrete Variable

• A probability distribution for a discrete variable is a mutually exclusive list of all possible numerical results for that variable along with the probability of occurrence of each outcome.

  • Example: Daily network outages

    • 0 days: 0.35

    • 1 day: 0.25

    • 2 days: 0.20

    • 3 days: 0.10

    • 4 days: 0.05

    • 5 days: 0.05

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Graphical Representation of Probability Distributions

• Probability distributions are often represented graphically.

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Expected Value of a Discrete Variable (Mean of Distribution, μ)

• The expected value (or mean) of a discrete variable (Weighted average)

  Daily Outages (xi)	Probability P(X=xi)	xiP(X=xi)
  0	0.35	0.00
  1	0.25	0.25
  2	0.20	0.40
  3	0.10	0.30
  4	0.05	0.20
  5	0.05	0.25
  Total	1.00	1.40

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Variance of a Discrete Variable

• Variance measures the dispersion of a discrete variable.

• Standard deviation of a discrete variable is calculated where:

  • E(X) = expected value of discrete variable X

  • xi = the i-th value of X

  • P(X=xi) = Probability of i-th occurrence of X

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Measuring Variance of Discrete Variables

• Daily network outages:

  Daily Outages (xi)	Probability P(X=xi)	[xi - E(X)]²	[xi - E(X)]²P(X=xi)
  0	0.35	1.96	0.686
  1	0.25	0.16	0.040
  2	0.20	0.36	0.072
  3	0.10	2.56	0.256
  4	0.05	6.76	0.338
  5	0.05	12.96	0.648

• Variance = σ² = 2.04, Standard Deviation = σ = 1.4283

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Covariance

• Covariance measures the strength of the relationship between two discrete variables X and Y.

• Positive covariance indicates a positive relationship.

• Negative covariance indicates a negative relationship.

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Mathematical Formula for Covariance

• Covariance formula:

  • where:

    • X = discrete variable X

    • xi = the i-th value of X

    • Y = discrete variable Y

    • yi = the i-th value of Y

    • P(X=xi,Y=yi) = probability of simultaneous occurrence of the i-th value of X and the i-th value of Y

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Investment Returns: Average Return

• Consider the returns of two investments of $1000 each under three different economic conditions.

  Economic Conditions	Probability	Investment A	Investment B
  Recession	0.2	-$25	-$200
  Stable Economy	0.5	+$50	+$60
  Growing Economy	0.3	+$100	+$350

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Investment Returns: Covariance

• Expected returns for each investment:

  • E(X) = μX = (-25)(.2) + (50)(.5) + (100)(.3) = 50

  • E(Y) = μY = (-200)(.2) + (60)(.5) + (350)(.3) = 95

  • Interpretation: Fund A has an average return of $50, and Fund B has an average return of $95 for each $1,000 investment.

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Investment Returns: Standard Deviation

• Interpretation: Although Fund B has a higher average return, it possesses greater volatility, thus increasing the chance of losses.

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Investment Returns: Covariance Interpretation

• Given that covariance is large and positive, there is a positive relationship between the two mutual funds, suggesting both are likely to increase or decrease together.

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Sum of Two Variables

• The expected value of the sum of two variables:

• The variance of the sum of two variables:

• The standard deviation of the sum of two variables:

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Expected Portfolio Return and Expected Risk

• Investment portfolios usually comprise several different combinations of funds (variables).

• Expected return and standard deviation can be computed simultaneously for two mutual funds.

• Investment Goal: Maximize average return while minimizing risk (standard deviation).

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Expected Portfolio Return and Risk

• Expected portfolio return (weighted average return):

• Portfolio risk (weighted volatility); where w = proportion of investment X in the portfolio value, (1 - w) = proportion of investment Y in the portfolio value.

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Portfolio Example

• Investment X: μX = 50 σX = 43.30

• Investment Y: μY = 95 σY = 193.21

• σXY = 8250

• If investment X constitutes 40% of the portfolio and Y 60%, portfolio return and risk (volatility) are computed based on their respective values.

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Continuous Probability Distributions

• Binomial

• Hypergeometric

• Poisson

• Discrete Probability Distributions

• Normal

• Uniform

• Exponential

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Binomial Probability Distribution

• Fixed number of observations, n (e.g., 15 coin tosses; 10 light bulbs from a warehouse).

• Each observation is classified as whether the "event of interest" occurred (e.g., heads or tails on each flip).

• The probability of an observation belonging to the category of the event of interest is denoted p; the probability that the event does not occur is (1 - p).

• The probability of an event occurring (p) is constant across all observations.

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Binomial Probability Distribution (Continued)

• Observations are independent.

• The outcome of one observation does not affect the outcome of another.

• Two sampling methods provide independence:

  • Infinite population without replacement

  • Finite population with replacement

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Possible Applications of the Binomial Distribution

• An industrial unit characterized products as defective or acceptable.

• A company bidding for contracts will either accept a contract or not.

• A market research company receives responses to the survey "yes, I will buy" or "no, I will not buy".

• New job applicants either accept the offer or reject it.

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Counting Techniques in Binomial Distribution

• Assuming the event of interest is obtaining heads when tossing a fair coin. Toss the coin three times.

• How many different ways can you achieve two heads? - Possible Ways: HHT, HTH, THH, thus there are three ways to get two heads.

• This case is simple. We need to be able to count ways for more complex situations.

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Counting Techniques in Binomial Distribution

• The number of combinations (combinations) choosing x elements from a population of n elements is given by the equation:

  • n! = (n)(n - 1)(n - 2) ... (2)(1)

  • x! = (x)(x - 1)(x - 2) ... (2)(1)

  • 0! = 1 (by definition)

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Counting Techniques: Combination Rule

• How many possible combinations of 3 ice cream scoops can you create at an ice cream shop if you can choose from 31 flavors and no flavor can be used more than once among the 3 scoops?

• Total choices are n = 31, and we are choosing x = 3.

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Binomial Distribution Equation

• P(X=x|n,p) = probability of x successes in n observations with probability p for each trial.

  • x = number of occurrences of the event of interest in the sample, (x = 0, 1, 2, ..., n)

  • n = number of observations (sample size)

  • p = probability of occurrence of the event of interest

• Example: Suppose x = # occurrences of "heads" in a coin toss: n = 4 p = 0.5 (1 - p) = (1 - 0.5) = 0.5 x = 0, 1, 2, 3, 4

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Example of Calculating a Binomial Probability

• What is the probability of one success in five trials if the probability of the event is 0.1? x = 1, n = 5, and p = 0.1

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Example of the Binomial Distribution

• Suppose the probability of buying a defective computer is 0.02. What is the probability of buying two defective computers from a sample of 10 computers? x = 2, n = 10, and p = 0.02

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Shape of the Binomial Distribution

• The shape of the binomial distribution depends on the values of p and n.

• If n = 5 and p = 0.1

• If n = 5 and p = 0.5

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Binomial Distribution Using Binomial Tables

• n = 10

  • p=0.20, p=0.25, p=0.30, p=0.35, p=0.40, p=0.45, p=0.50

• Examples: n = 10, p = 0.35, x = 3: P(X=3|10,0.35) = 0.2522

• n = 10, p = 0.75, x = 8: P(X=8|10,0.75) = 0.2816

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Characteristics of Binomial Distribution

• Mean, Variance, and Standard Deviation where

  • n = sample size

  • p = probability of occurrence of the event in each trial

  • (1 - p) = probability of not occurring of the event in each trial

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Characteristics of Binomial Distribution

• Various shapes of the binomial distribution are observed.

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Tools for Computing Binomial Distribution

• Both Excel and Minitab can be used to compute binomial distributions.

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Poisson Distribution: Definitions

• Use the Poisson distribution when interested in the number of times an event occurs in a given opportunity area.

• An opportunity area is a continuous unit or time interval, volume, or area where more than one event may occur.

• Example: Number of scratches in a car paint, number of mosquito bites on a person, number of non-responses by a computer in a day.

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Poisson Distribution

• Apply Poisson distribution when:

  • You want to measure how many times an event occurs in a given opportunity area.

  • The probability of an event occurring in an opportunity area is the same for all opportunity areas.

  • The number of events occurring in one opportunity area is independent of the number of events occurring in any other area.

  • The probability of two or more events occurring in a single opportunity area approaches zero as the area becomes smaller.

  • The average number of events per unit is λ (lambda).

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Poisson Distribution Equation

• Where:

  • x = number of events in an opportunity area

  • λ = expected number of events

  • e = base of the natural logarithm (2.71828...)

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Characteristics of Poisson Distribution

• Mean, Variance, and Standard Deviation where

  • λ = expected number of events

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Use of Poisson Tables

• (Available Online)

• Example: Find P(X = 2 | λ = 0.50)

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Tools for Computing Poisson Distribution

• Both Excel and Minitab can be used for Poisson distribution computations.

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Poisson Probability Graph

• When λ = 0.50

• When λ = 3.00

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Shape of Poisson Distribution

• The shape of the Poisson distribution depends on the parameter λ.

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Hypergeometric Distribution

• Binomial distribution applies when the sample is chosen with replacement from an infinite population or without replacement from a finite population.

• Hypergeometric distribution applies when sampling without replacement from a finite population.

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Hypergeometric Distribution

• Sample size "n" from a finite population size "N"

• The sampling is done without replacement.

• The results of observations are dependent.

• Interested in finding the probability of occurrence of the event of interest X times in the sample when there are "E" events of interest in the population.

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Hypergeometric Distribution Equation

• Where

  • N = population size

  • E = number of events of interest in the population

  • N - E = number of non-events of interest in the population

  • n = sample size

  • x = number of events of interest in the sample

  • n - x = number of non-events of interest in the sample

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Properties of Hypergeometric Distribution

• The mean of the hypergeometric distribution is:

• The standard deviation is:

• Where the correction factor of the finite population occurs due to sampling without replacement from a finite population.

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Use of Hypergeometric Distribution

• Example: Examining 3 different computers from a population of 10 computers. - 4 out of the 10 computers use illegal software. - What is the probability that 2 out of the 3 selected computers use illegal software?

  • N = 10, n = 3, A = 4, x = 2

  • Probability that 2 out of the 3 selected computers use illegal software is 0.30, or 30%.

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Tools for Computing Hypergeometric Distribution

• Both Excel and Minitab can be used for hypergeometric distribution calculations.

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Chapter 5 Online Topic

• The use of Poisson distribution as an approximation of the binomial distribution can be found online.

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Chapter Summary

In this chapter, we covered:

• The probability distribution of a discrete variable

• Covariance and its application in finance

• The binomial distribution

• The hypergeometric distribution

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Poisson Distribution as an Approximation of Binomial Distribution

• Online Topic Chapter 5

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Learning Objective

• Understand the application of Poisson distribution as an approximation of the binomial distribution.

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Application of Poisson Distribution as an Approximation of Binomial Distribution

• The binomial distribution is discrete while the normal distribution is continuous.

• The use of the normal distribution as an approximation to the binomial distribution improves accuracy if continuity correction is applied.

  • Example: If X is discrete in a binomial distribution, P(X = 4 | n, p) can be approximated with a continuous normal distribution by finding P(3.5 < X < 4.5).

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Application of Poisson Distribution as an Approximation of Binomial Distribution

• As p approaches 0.5, the approximation of Poisson to binomial improves.

• The larger the sample size n, the better the approximation of Poisson to binomial.

• General rule: The normal distribution can be used to approximate the binomial if nP ≥ 5 and n(1 - P) ≥ 5 (continuation).

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Application of Poisson Distribution as an Approximation of Binomial Distribution

• The mean and standard deviation of the binomial distribution are as follows:

  • μ = nP

• The binomial is transformed into normal by the formula: (continuation).

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Application of Poisson Distribution as an Approximation of Binomial Distribution

• If n = 1000 and p = 0.2, P(X ≤ 180)?

  • Approximate P(X ≤ 180 | 1000, 0.2) using continuity correction: P(X ≤ 180.5)

  • Transform to standard normal distribution:

  • P(Z ≤ -1.54) = 0.0618.

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Summary

• Finding approximations of probability distributions using the normal distribution.