Chapter 6 Discrete Probability Distributions
description
Transcript of Chapter 6 Discrete Probability Distributions
Chapter 6
Discrete
Probability Distributions
BUS210: Business Statistics Discrete Probability Distributions - 2
Learning Objectives
In this section, you will learn about: Probability distributions and their properties Important discrete probability distributions:
The Binomial Distribution The Poisson Distribution
Calculating the expected value and variance Using the Binomial and Poisson distributions
to solve business problems
BUS210: Business Statistics Discrete Probability Distributions - 3
Definitions
Random variable: representation of a possible numerical value from an uncertain event. Discrete random variables: “How many”
produce outcomes that come from counting (e.g. number of courses you are taking).
Continuous random variables: “How much”
produce outcomes that come from a measurement (e.g. your annual salary, or your weight).
BUS210: Business Statistics Discrete Probability Distributions - 4
Random
Variables
Discrete Random Variable
ContinuousRandom Variable
Ch. 6 Ch. 7
Random Variables
Discrete Random Variable
BUS210: Business Statistics Discrete Probability Distributions - 5
Can only assume a countable number of values
Discrete Random Variables
If X is the number of times that a 6 shows up, then X could be equal to 0, 1, 2, 3, 4, or 5
If X is the number of times that heads comes up, then X could be equal to 0, 1, 2, or 3
Flipping a coin 3 times.
Examples:
Rolling five dice (as in Yahtzee)
BUS210: Business Statistics Discrete Probability Distributions - 6
A probability distribution is…
a mutually exclusive listing of all possible numerical outcomes for a variable and the probability of an occurrence associated with each outcome.
Number of Classes Taken Probability
1 0.20
2 0.10
3 0.40
4 0.24
5 or more 0.06
Discrete Random Variables
Notice that the total probability adds up to 1.00
Example for a discrete random variable:
BUS210: Business Statistics Discrete Probability Distributions - 7
X Freq Probability
0 1 1/4 = 0.25
1 2 2/4 = 0.50
2 1 1/4 = 0.25
Experiment: Toss 2 Coins. Let X = # heads.
Probability Distribution
0 1 2 X
0.50
0.25
Pro
bab
ility
Discrete Random Variable
Probability Distribution
T T
T H
TH
H HTotal = 1.00
BUS210: Business Statistics Discrete Probability Distributions - 8
Example: Experiment of 2 Tossed Coins
Expected Value is the mean of a discrete random variable
Discrete Random Variable
Expected Value
i X P(X)X1 0 1/4 = 0.25
X2 1 2/4 = 0.50
X3 2 1/4 = 0.25
Also known as the Weighted Average
XiP(Xi)
(0)(0.25) = 0.00
(1)(0.50) = 0.50
(2)(0.25) = 0.50
E(X) = 1.00
BUS210: Business Statistics Discrete Probability Distributions - 9
Variance of a discrete random variable
Standard Deviation of a discrete random variable
Discrete Random Variable
Measuring Dispersion
BUS210: Business Statistics Discrete Probability Distributions - 10
Example: standard deviation of 2 tossed coins
(continued)
Discrete Random Variable
Measuring Dispersion
X E(X) X-E(X) (X-E(X))2 P(X) (X-E(X))2P(X)
X1 0 1 -1 1 0.25 0.25
X2 1 1 0 0 0.50 0.00
X3 2 1 +1 1 0.25 0.25
∑= 0.50
BUS210: Business Statistics Discrete Probability Distributions - 11
Continuous Probability
Distributions
Probability Distributions
Binomial
Poisson
Discrete Probability
Distributions
Normal
Ch. 6 Ch. 7
Probability Distributions
Binomial
Exponential
BUS210: Business Statistics Discrete Probability Distributions - 12
A finite number of observations, n 20 tosses of a coin 12 auto batteries purchased from a wholesaler
Each observation is categorized as… Success: “event of interest” has occurred, or
i.e. - heads (or tails) in each toss of a coin; i.e. - defective (or not defective) battery
Failure: “event of interest” has not occurred The complement of a success
These two categories, success & failure, are mutually exclusive and collectively exhaustive
Probability Distributions
Binomial
Note: Note: A random A random experiment with experiment with only 2 outcomes is only 2 outcomes is known as a known as a Bernoulli Bernoulli experimentexperiment..
BUS210: Business Statistics Discrete Probability Distributions - 13
(continued)
The probability of… success (event occurring) is represented as π failure (event not occurring) is 1 – π
Probability of success (π) is constant Probability of getting tails is the same for each toss of the coin
Observations are independent Each event is unaffected by any other event Two sampling methods deliver independence
Infinite population without replacement Finite population with replacement
Probability Distributions
Binomial
BUS210: Business Statistics Discrete Probability Distributions - 14
Applications: Situations where are there only two outcomes
Apple marks newly manufactured iPads as either defective or acceptable
Visitors to Amazon’s website will either buy an item or not buy an item.
Asking if voters will approve a referendum, a pollster receives responses of “yes” or “no”
Federal Express marks a delivery as either damaged or not damaged
Probability Distributions
Binomial (continued)
BUS210: Business Statistics Discrete Probability Distributions - 15
You toss a coin three times.
In how many ways can you get two heads?
Probability Distributions
Binomial (continued)
Possible ways: HHT, HTH, THH, so there are three ways you can getting two heads.
This situation is fairly simple….but, what about 10 times?…a 100?...or even a thousand?
Counting Revisited:
For more complicated situations we need a methodto be able to count the number of ways (combinations).
BUS210: Business Statistics Discrete Probability Distributions - 16
The number of combinations of
selecting x objects out of n objects is
where: n! means “n factorial”2! = (1)(2)4! = (1)(2)(3)(4)
Note: 0! = 1 (by definition)
Counting Techniques
Rule of Combinations
BUS210: Business Statistics Discrete Probability Distributions - 17
combinations
Counting Techniques
Rule of Combinations
You visit Baskin & Robbins. How many different possible 3 scoop combinations could you decide on for your ice cream cone if you select from their 31 flavors?
The total choices is n = 31, and we select x = 3.
BUS210: Business Statistics Discrete Probability Distributions - 18
π = probability of “event of interest”
n = sample size (number of trials or observations)x = number of “events of interest” in sample
Probability Distributions
Binomial
n!(n-x)!n!
P(x) = π x (1-π) n-x
Note: other sources may show this equation as or similar.
for the number of heads in 3 coin flips, we would use…
π = 0.5 and (1 – π) = 0.5
n = 4
X = can be 0, or 1, or 2, or 3
Each value of X will have a different probability
BUS210: Business Statistics Discrete Probability Distributions - 19
What is the probability of 2 successes in 7 observations if the probability of the event of interest is 0.4?
Probability Distributions
Binomial Example
x = 1, n = 7, and π = 0.4
BUS210: Business Statistics Discrete Probability Distributions - 20
The probability of a single customer purchasing an extended warranty is 0.35.
What is the probability of having 3 of the next 10 customers purchase an extended warranty?
Probability Distributions
Binomial Example
BUS210: Business Statistics Discrete Probability Distributions - 21
n x … π=.20 π=.25 π=.30 π=.35 π=.40 π=.45 π=.50
10 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
Examples:
Probability Distributions
Using the Binomial Table
n = 10, π = .35, x = 3
n = 10, π = .75, x = 2
P(x = 3) = .2522
P(x = 2) = .0004
BUS210: Business Statistics Discrete Probability Distributions - 22
The shape of the binomial distribution…
depends on the values of π and n
Probability Distributions
Shape of the Binomial
n = 5 π = 0.5n = 5 π = 0.1
0 1 2 3 4 5X
P(X)
.2
.4
.6
0
0 1 2 3 4 5X
P(X)
.2
.4
.6
0
BUS210: Business Statistics Discrete Probability Distributions - 23
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
BUS210: Business Statistics Discrete Probability Distributions - 24
n = 5 π = 0.1
n = 5 π = 0.5
Examples
Binomial Distribution Characteristics
0 1 2 3 4 5X
P(X)
.2
.4
.6
0
0 1 2 3 4 5X
P(X)
.2
.4
.6
0
BUS210: Business Statistics Discrete Probability Distributions - 25
ExamplesExamples: : •The probability that The probability that less than less than 3 of the next 10 customers will purchase an 3 of the next 10 customers will purchase an extended warranty isextended warranty is
Binomial Distribution Characteristics
Compound EventsCompound Events
P(x<3) = P(x=0) + P(x=1) + P(x=2)
P(x<3) = P(x=0) + P(x=1) + P(x=2) + P(x=3)
Individual probabilities can be added to obtain any desired combined Individual probabilities can be added to obtain any desired combined event probability.event probability.
• The probability that 3 The probability that 3 or fewer or fewer of the next 10 customers will purchase an of the next 10 customers will purchase an extended warranty isextended warranty is
= 0.0135 + 0.0725 + 0.1757 = 0.2617
= 0.0135 + 0.0725 + 0.1757 + 0.2522 = 0.5139
BUS210: Business Statistics Discrete Probability Distributions - 26
Binomial DistributionUsing Excel
Cell Formulas
n x π
x n π cum
n x π x (1-π)
BUS210: Business Statistics Discrete Probability Distributions - 27
Continuous Probability
Distributions
Binomial
Poisson
Probability Distributions
Discrete Probability
Distributions
Normal
Ch. 6 Ch. 7
Probability Distributions
Poisson Exponential
BUS210: Business Statistics Discrete Probability Distributions - 28
Used when you are interested in…the number of times an event occurswithin a continuous unit or interval.
Such as time, distance, volume, or area.
Examples: The number of… potholes in mile of road (distance) computer crashes in a day (time) chocolate chips in a cookie (volume) mosquito bites on a person (area)
Probability Distributions
The Poisson Distribution
Was actually Was actually used in a study used in a study of malariaof malaria
BUS210: Business Statistics Discrete Probability Distributions - 29
Defining features: Independent: The occurrence of any single event
does not impact the occurrence of any other event. Constant: The average probability of an event
occurring in one area of opportunity remains the same for all other similar areas. The average number of events per unit is represented by (lambda)
The average probability of an event decreases or increases proportionally to any decrease or increase in the size of the area of opportunity.
Probability Distributions
The Poisson Distribution
BUS210: Business Statistics Discrete Probability Distributions - 30
where:
x = number of events in an area of opportunity
= expected value (mean) of number of events
e = base of the natural logarithm system (2.71828...)
Probability Distributions
The Poisson Distribution
Note: other sources may show this equation as or similar.
e -λ λx P(x) =
x!
BUS210: Business Statistics Discrete Probability Distributions - 31
Mean
Variance and Standard Deviation
where = expected number of events
Poisson DistributionCharacteristics
Isn’t that interesting?
BUS210: Business Statistics Discrete Probability Distributions - 32
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
Example: Find P(X = 2) if = 0.50
Probability Distributions
Using the Poisson Table
BUS210: Business Statistics Discrete Probability Distributions - 33
Poisson DistributionUsing Excel
Cell Formulas
x λ cum
BUS210: Business Statistics Discrete Probability Distributions - 34
X P(X)
0
1
2
3
4
5
6
7
0.6065
0.3033
0.0758
0.0126
0.0016
0.0002
0.0000
0.0000
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0 1 2 3 4 5 6 7
x
P(x
)
P(X=2) = 0.0758
= 0.50
Probability Distributions
Graphing the Poisson
BUS210: Business Statistics Discrete Probability Distributions - 35
The shape of the Poisson Distribution depends on the parameter :
0.00
0.05
0.10
0.15
0.20
0.25
1 2 3 4 5 6 7 8 9 10 11 12
x
P(x
)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0 1 2 3 4 5 6 7
x
P(x
)
= 0.50 = 3.00
Probability Distributions
Shape of the Poisson
BUS210: Business Statistics Discrete Probability Distributions - 36
Chapter Summary
Probability distributions Discrete random variables The Binomial distribution The Poisson distribution