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levine_smume7_ch051.pptStatistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Chapter 5
Discrete Probability Distributions
Statistics for Managers Using Microsoft Excel
7th Edition
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
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 the binomial, hypergeometric, and Poisson distributions can be used to solve business problems
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Definitions
- Discrete variables produce outcomes that come from a counting process (e.g. number of classes you are taking).
- Continuous variables produce outcomes that come from a measurement (e.g. your annual salary, or your weight).
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Types Of Variables
Discrete
Random Variable
Continuous
Random Variable
Ch. 5
Ch. 6
Discrete
Random Variable
Continuous
Random Variable
Ch. 5
Ch. 6
Types Of
Variables
Discrete
Variable
Continuous
Variable
Ch. 5
Ch. 6
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Discrete Random Variables
- Can only assume a countable number of values
Examples:
- Roll a die twice
Let X be the number of times 4 occurs
(then X could be 0, 1, or 2 times)
- Toss a coin 5 times.
Let X be the number of heads
(then X = 0, 1, 2, 3, 4, or 5)
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Probability Distribution For A Discrete Random Variable
- A probability distribution for a discrete random variable is a mutually exclusive listing of all possible numerical outcomes for that variable and a probability of occurrence associated with each outcome.
| Number of Classes Taken | Probability |
| 2 | 0.20 |
| 3 | 0.40 |
| 4 | 0.24 |
| 5 | 0.16 |
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Experiment: Toss 2 Coins. Let X = # heads.
T
T
Example of a Discrete Random Variable Probability Distribution
4 possible outcomes
T
T
H
H
H
H
Probability Distribution
0 1 2 X
X Value Probability
0 1/4 = 0.25
1 2/4 = 0.50
2 1/4 = 0.25
0.50
0.25
Probability
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Discrete Variables
Expected Value (Measuring Center)
- Expected Value (or mean) of a discrete
variable (Weighted Average)
- Example: Toss 2 coins,
X = # of heads,
compute expected value of X:
E(X) = ((0)(0.25) + (1)(0.50) + (2)(0.25))
= 1.0
X P(X=Xi)
0 0.25
1 0.50
2 0.25
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
- Variance of a discrete random variable
- Standard Deviation of a discrete random variable
where:
E(X) = Expected value of the discrete random variable X
Xi = the ith outcome of X
P(X=Xi) = Probability of the ith occurrence of X
Discrete Random Variables
Measuring Dispersion
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
- Example: Toss 2 coins, X = # heads,
compute standard deviation (recall E(X) = 1)
Discrete Random Variables
Measuring Dispersion
(continued)
Possible number of heads = 0, 1, or 2
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Covariance
- The covariance measures the strength of the linear relationship between two discrete random variables X and Y.
- A positive covariance indicates a positive relationship.
- A negative covariance indicates a negative relationship.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
The Covariance Formula
- The covariance formula:
where: X = discrete random variable X
Xi = the ith outcome of X
Y = discrete random variable Y
Yi = the ith outcome of Y
P(X=Xi,Y=Yi) = probability of occurrence of the
ith outcome of X and the ith outcome of Y
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Investment Returns
The Mean
Consider the return per $1000 for two types of investments.
| Economic Condition Prob. | Investment | |
| Passive Fund X | Aggressive Fund Y | |
| 0.2 Recession | - $25 | - $200 |
| 0.5 Stable Economy | + $50 | + $60 |
| 0.3 Expanding Economy | + $100 | + $350 |
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Investment Returns
The Mean
E(X) = μX = (-25)(.2) +(50)(.5) + (100)(.3) = 50
E(Y) = μY = (-200)(.2) +(60)(.5) + (350)(.3) = 95
Interpretation: Fund X is averaging a $50.00 return and fund Y is averaging a $95.00 return per $1000 invested.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Investment Returns
Standard Deviation
Interpretation: Even though fund Y has a higher average return, it is subject to much more variability and the probability of loss is higher.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Investment Returns
Covariance
Interpretation: Since the covariance is large and positive, there is a positive relationship between the two investment funds, meaning that they will likely rise and fall together.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
The Sum of
Two Random Variables
- Expected Value of the sum of two random variables:
- Variance of the sum of two random variables:
- Standard deviation of the sum of two random variables:
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Portfolio Expected Return and Expected Risk
- Investment portfolios usually contain several different funds (random variables)
- The expected return and standard deviation of two funds together can now be calculated.
- Investment Objective: Maximize return (mean) while minimizing risk (standard deviation).
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Portfolio Expected Return
and Portfolio Risk
- Portfolio expected return (weighted average return):
- Portfolio risk (weighted variability)
Where w = proportion of portfolio value in asset X
(1 - w) = proportion of portfolio value in asset Y
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Portfolio Example
Investment X: μX = 50 σX = 43.30
Investment Y: μY = 95 σY = 193.21
σXY = 8250
Suppose 40% of the portfolio is in Investment X and 60% is in Investment Y:
The portfolio return and portfolio variability are between the values for investments X and Y considered individually
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Probability Distributions
Continuous
Probability Distributions
Binomial
Hypergeometric
Poisson
Probability Distributions
Discrete
Probability Distributions
Normal
Uniform
Exponential
Ch. 5
Ch. 6
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 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
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Binomial Probability Distribution
(continued)
- Observations are independent
- The outcome of one observation does not affect the outcome of the other
- Two sampling methods deliver independence
- Infinite population without replacement
- Finite population with replacement
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
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
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
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.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Counting Techniques
Rule of Combinations
- The number of combinations of selecting X objects out of n objects is
where:
n! =(n)(n - 1)(n - 2) . . . (2)(1)
X! = (X)(X - 1)(X - 2) . . . (2)(1)
0! = 1 (by definition)
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
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.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
P(X=x|n,π) = probability of x events of interest in n trials, with the probability of an “event of interest” being π for each trial
x = number of “events of interest” in sample,
(x = 0, 1, 2, ..., n)
n = sample size (number of trials
or observations)
π = probability of “event of interest”
P(X=x |n,π)
n
x!
n
x
π
(1-π)
x
n
x
!
(
)
!
=
-
-
Example: Flip a coin four times, let x = # heads:
n = 4
π = 0.5
1 - π = (1 - 0.5) = 0.5
X = 0, 1, 2, 3, 4
Binomial Distribution Formula
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Example:
Calculating a Binomial Probability
What is the probability of one success in five observations if the probability of an event of interest is 0.1?
x = 1, n = 5, and π = 0.1
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
The Binomial Distribution
Example
Suppose the probability of purchasing a defective computer is 0.02. What is the probability of purchasing 2 defective computers in a group of 10?
x = 2, n = 10, and π = 0.02
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
The Binomial Distribution
Shape
0
.2
.4
.6
0
1
2
3
4
5
x
P(X=x|5, 0.1)
.2
.4
.6
0
1
2
3
4
5
x
P(X=x|5, 0.5)
0
- The shape of the binomial distribution depends on the values of π and n
- Here, n = 5 and π = .1
- Here, n = 5 and π = .5
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
The Binomial Distribution Using Binomial Tables (Available On Line)
Examples:
n = 10, π = 0.35, x = 3: P(X = 3|10, 0.35) = 0.2522
n = 10, π = 0.75, x = 8: P(X = 8|10, 0.75) = 0.0004
| n = 10 | |||||||||
| x | … | π=.20 | π=.25 | π=.30 | π=.35 | π=.40 | π=.45 | π=.50 | |
| 0 1 2 3 4 5 6 7 8 9 10 | … … … … … … … … … … … | 0.1074 0.2684 0.3020 0.2013 0.0881 0.0264 0.0055 0.0008 0.0001 0.0000 0.0000 | 0.0563 0.1877 0.2816 0.2503 0.1460 0.0584 0.0162 0.0031 0.0004 0.0000 0.0000 | 0.0282 0.1211 0.2335 0.2668 0.2001 0.1029 0.0368 0.0090 0.0014 0.0001 0.0000 | 0.0135 0.0725 0.1757 0.2522 0.2377 0.1536 0.0689 0.0212 0.0043 0.0005 0.0000 | 0.0060 0.0403 0.1209 0.2150 0.2508 0.2007 0.1115 0.0425 0.0106 0.0016 0.0001 | 0.0025 0.0207 0.0763 0.1665 0.2384 0.2340 0.1596 0.0746 0.0229 0.0042 0.0003 | 0.0010 0.0098 0.0439 0.1172 0.2051 0.2461 0.2051 0.1172 0.0439 0.0098 0.0010 | 10 9 8 7 6 5 4 3 2 1 0 |
| … | π=.80 | π=.75 | π=.70 | π=.65 | π=.60 | π=.55 | π=.50 | x |
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Binomial Distribution Characteristics
- Mean
- Variance and Standard Deviation
Where n = sample size
π = probability of the event of interest for any trial
(1 – π) = probability of no event of interest for any trial
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
The Binomial Distribution
Characteristics
0
.2
.4
.6
0
1
2
3
4
5
x
P(X=x|5, 0.1)
.2
.4
.6
0
1
2
3
4
5
x
P(X=x|5, 0.5)
0
Examples
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Using Excel For The
Binomial Distribution
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
The Poisson Distribution
Definitions
- You use the Poisson distribution when you are interested in the number of times an event occurs in a given area of opportunity.
- An area of opportunity is a continuous unit or interval of time, volume, or such area in which more than one occurrence of an event can occur.
- The number of scratches in a car’s paint
- The number of mosquito bites on a person
- The number of computer crashes in a day
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
The Poisson Distribution
- Apply the Poisson Distribution when:
- You wish to count the number of times an event occurs in a given area of opportunity
- The probability that an event occurs in one area of opportunity is the same for all areas of opportunity
- The number of events that occur in one area of opportunity is independent of the number of events that occur in the other areas of opportunity
- The probability that two or more events occur in an area of opportunity approaches zero as the area of opportunity becomes smaller
- The average number of events per unit is (lambda)
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Poisson Distribution Formula
where:
x = number of events in an area of opportunity
= expected number of events
e = base of the natural logarithm system (2.71828...)
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Poisson Distribution Characteristics
- Mean
- Variance and Standard Deviation
where = expected number of events
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Using Poisson Tables (Available On Line)
Example: Find P(X = 2 | = 0.50)
| X | | ||||||||
| 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 | 0.90 | |
| 0 1 2 3 4 5 6 7 | 0.9048 0.0905 0.0045 0.0002 0.0000 0.0000 0.0000 0.0000 | 0.8187 0.1637 0.0164 0.0011 0.0001 0.0000 0.0000 0.0000 | 0.7408 0.2222 0.0333 0.0033 0.0003 0.0000 0.0000 0.0000 | 0.6703 0.2681 0.0536 0.0072 0.0007 0.0001 0.0000 0.0000 | 0.6065 0.3033 0.0758 0.0126 0.0016 0.0002 0.0000 0.0000 | 0.5488 0.3293 0.0988 0.0198 0.0030 0.0004 0.0000 0.0000 | 0.4966 0.3476 0.1217 0.0284 0.0050 0.0007 0.0001 0.0000 | 0.4493 0.3595 0.1438 0.0383 0.0077 0.0012 0.0002 0.0000 | 0.4066 0.3659 0.1647 0.0494 0.0111 0.0020 0.0003 0.0000 |
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Using Excel For The
Poisson Distribution
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Graph of Poisson Probabilities
P(X = 2 | =0.50) = 0.0758
Graphically:
= 0.50
| X | = 0.50 |
| 0 1 2 3 4 5 6 7 | 0.6065 0.3033 0.0758 0.0126 0.0016 0.0002 0.0000 0.0000 |
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chart1
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Histogram
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Poisson2
| Poisson Probabilities for Customer Arrivals | ||||||
| Data | ||||||
| Average/Expected number of successes: | 0.5 | |||||
| Poisson Probabilities Table | ||||||
| X | P(X) | P(<=X) | P(<X) | P(>X) | P(>=X) | |
| 0 | 0.606531 | 0.606531 | 0.000000 | 0.393469 | 1.000000 | |
| 1 | 0.303265 | 0.909796 | 0.606531 | 0.090204 | 0.393469 | |
| 2 | 0.075816 | 0.985612 | 0.909796 | 0.014388 | 0.090204 | |
| 3 | 0.012636 | 0.998248 | 0.985612 | 0.001752 | 0.014388 | |
| 4 | 0.001580 | 0.999828 | 0.998248 | 0.000172 | 0.001752 | |
| 5 | 0.000158 | 0.999986 | 0.999828 | 0.000014 | 0.000172 | |
| 6 | 0.000013 | 0.999999 | 0.999986 | 0.000001 | 0.000014 | |
| 7 | 0.000001 | 1.000000 | 0.999999 | 0.000000 | 0.000001 | |
| 8 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 9 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 10 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 11 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 12 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 13 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 14 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 15 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 16 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 17 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 18 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 19 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 20 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 |
Poisson2
Poisson
| Poisson Probabilities for Customer Arrivals | ||||||
| Data | ||||||
| Average/Expected number of successes: | 0.1 | |||||
| Poisson Probabilities Table | ||||||
| X | P(X) | P(<=X) | P(<X) | P(>X) | P(>=X) | |
| 0 | 0.9048 | 0.904837 | 0.000000 | 0.095163 | 1.000000 | |
| 1 | 0.0905 | 0.995321 | 0.904837 | 0.004679 | 0.095163 | |
| 2 | 0.0045 | 0.999845 | 0.995321 | 0.000155 | 0.004679 | |
| 3 | 0.0002 | 0.999996 | 0.999845 | 0.000004 | 0.000155 | |
| 4 | 0.0000 | 1.000000 | 0.999996 | 0.000000 | 0.000004 | |
| 5 | 0.0000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 6 | 0.0000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 7 | 0.0000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 |
Sheet1
Sheet2
Sheet3
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Poisson Distribution Shape
- The shape of the Poisson Distribution depends on the parameter :
= 0.50
= 3.00
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chart1
| 0.0497870684 |
| 0.1493612051 |
| 0.2240418077 |
| 0.2240418077 |
| 0.1680313557 |
| 0.1008188134 |
| 0.0504094067 |
| 0.0216040315 |
| 0.0081015118 |
| 0.0027005039 |
| 0.0008101512 |
| 0.0002209503 |
Histogram
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Poisson2
| Poisson Probabilities for Customer Arrivals | ||||||
| Data | ||||||
| Average/Expected number of successes: | 3 | |||||
| Poisson Probabilities Table | ||||||
| X | P(X) | P(<=X) | P(<X) | P(>X) | P(>=X) | |
| 0 | 0.049787 | 0.049787 | 0.000000 | 0.950213 | 1.000000 | |
| 1 | 0.149361 | 0.199148 | 0.049787 | 0.800852 | 0.950213 | |
| 2 | 0.224042 | 0.423190 | 0.199148 | 0.576810 | 0.800852 | |
| 3 | 0.224042 | 0.647232 | 0.423190 | 0.352768 | 0.576810 | |
| 4 | 0.168031 | 0.815263 | 0.647232 | 0.184737 | 0.352768 | |
| 5 | 0.100819 | 0.916082 | 0.815263 | 0.083918 | 0.184737 | |
| 6 | 0.050409 | 0.966491 | 0.916082 | 0.033509 | 0.083918 | |
| 7 | 0.021604 | 0.988095 | 0.966491 | 0.011905 | 0.033509 | |
| 8 | 0.008102 | 0.996197 | 0.988095 | 0.003803 | 0.011905 | |
| 9 | 0.002701 | 0.998898 | 0.996197 | 0.001102 | 0.003803 | |
| 10 | 0.000810 | 0.999708 | 0.998898 | 0.000292 | 0.001102 | |
| 11 | 0.000221 | 0.999929 | 0.999708 | 0.000071 | 0.000292 | |
| 12 | 0.000055 | 0.999984 | 0.999929 | 0.000016 | 0.000071 | |
| 13 | 0.000013 | 0.999997 | 0.999984 | 0.000003 | 0.000016 | |
| 14 | 0.000003 | 0.999999 | 0.999997 | 0.000001 | 0.000003 | |
| 15 | 0.000001 | 1.000000 | 0.999999 | 0.000000 | 0.000001 | |
| 16 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 17 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 18 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 19 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 20 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 |
Poisson2
Poisson
Sheet1
| Poisson Probabilities for Customer Arrivals | ||||||
| Data | ||||||
| Average/Expected number of successes: | 0.1 | |||||
| Poisson Probabilities Table | ||||||
| X | P(X) | P(<=X) | P(<X) | P(>X) | P(>=X) | |
| 0 | 0.9048 | 0.904837 | 0.000000 | 0.095163 | 1.000000 | |
| 1 | 0.0905 | 0.995321 | 0.904837 | 0.004679 | 0.095163 | |
| 2 | 0.0045 | 0.999845 | 0.995321 | 0.000155 | 0.004679 | |
| 3 | 0.0002 | 0.999996 | 0.999845 | 0.000004 | 0.000155 | |
| 4 | 0.0000 | 1.000000 | 0.999996 | 0.000000 | 0.000004 | |
| 5 | 0.0000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 6 | 0.0000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 7 | 0.0000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 |
Sheet2
Sheet3
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Histogram
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Poisson2
| Poisson Probabilities for Customer Arrivals | ||||||
| Data | ||||||
| Average/Expected number of successes: | 0.5 | |||||
| Poisson Probabilities Table | ||||||
| X | P(X) | P(<=X) | P(<X) | P(>X) | P(>=X) | |
| 0 | 0.606531 | 0.606531 | 0.000000 | 0.393469 | 1.000000 | |
| 1 | 0.303265 | 0.909796 | 0.606531 | 0.090204 | 0.393469 | |
| 2 | 0.075816 | 0.985612 | 0.909796 | 0.014388 | 0.090204 | |
| 3 | 0.012636 | 0.998248 | 0.985612 | 0.001752 | 0.014388 | |
| 4 | 0.001580 | 0.999828 | 0.998248 | 0.000172 | 0.001752 | |
| 5 | 0.000158 | 0.999986 | 0.999828 | 0.000014 | 0.000172 | |
| 6 | 0.000013 | 0.999999 | 0.999986 | 0.000001 | 0.000014 | |
| 7 | 0.000001 | 1.000000 | 0.999999 | 0.000000 | 0.000001 | |
| 8 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 9 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 10 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 11 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 12 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 13 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 14 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 15 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 16 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 17 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 18 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 19 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 20 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 |
Poisson2
Poisson
| Poisson Probabilities for Customer Arrivals | ||||||
| Data | ||||||
| Average/Expected number of successes: | 0.1 | |||||
| Poisson Probabilities Table | ||||||
| X | P(X) | P(<=X) | P(<X) | P(>X) | P(>=X) | |
| 0 | 0.9048 | 0.904837 | 0.000000 | 0.095163 | 1.000000 | |
| 1 | 0.0905 | 0.995321 | 0.904837 | 0.004679 | 0.095163 | |
| 2 | 0.0045 | 0.999845 | 0.995321 | 0.000155 | 0.004679 | |
| 3 | 0.0002 | 0.999996 | 0.999845 | 0.000004 | 0.000155 | |
| 4 | 0.0000 | 1.000000 | 0.999996 | 0.000000 | 0.000004 | |
| 5 | 0.0000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 6 | 0.0000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | |
| 7 | 0.0000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 |
Sheet1
Sheet2
Sheet3
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
The Hypergeometric Distribution
- The binomial distribution is applicable when selecting from a finite population with replacement or from an infinite population without replacement.
- The hypergeometric distribution is applicable when selecting from a finite population without replacement.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
The Hypergeometric Distribution
- “n” trials in a sample taken from a finite population of size N
- Sample taken without replacement
- Outcomes of trials are dependent
- Concerned with finding the probability of “X” items of interest in the sample where there are “A” items of interest in the population
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Hypergeometric Distribution Formula
Where
N = population size
A = number of items of interest in the population
N – A = number of events not of interest in the population
n = sample size
x = number of items of interest in the sample
n – x = number of events not of interest in the sample
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Properties of the
Hypergeometric Distribution
- The mean of the hypergeometric distribution is
- The standard deviation is
Where is called the “Finite Population Correction Factor”
from sampling without replacement from a
finite population
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Using the
Hypergeometric Distribution
Example: 3 different computers are checked out from 10 in the department. 4 of the 10 computers have illegal software loaded. What is the probability that 2 of the 3 selected computers have illegal software loaded?
N = 10 n = 3
A = 4 x = 2
The probability that 2 of the 3 selected computers have illegal software loaded is 0.30, or 30%.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Using Excel for the
Hypergeometric Distribution
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
Chapter Summary
In this chapter we discussed
- The probability distribution of a discrete random variable
- The covariance and its application in finance
- The Binomial distribution
- The Poisson distribution
- The Hypergeometric distribution
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.
Chap 5-*
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.
Printed in the United States of America.
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