Lecture 8: Tue, Oct 1

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Lecture 8 Tue, Oct 1

• Announcements
• HW 3 due Thu
• HW 2 back on Thu
• Practice Exam 1 now posted
• Todays Material
• More on joint, conditional prob. trees
• Random Variables
• Discrete Probability Distributions
• Mean and variance of discrete RVs

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Example

• The Philadelphia 76ers win 70 of the time when
  Allen Iverson plays, and 40 of the time when he
  doesnt. Iverson plays in 85 of the 76ers
  games. (Solution using trees)
• a) What is the (marginal) probability that the
  76ers win?
• b) Given that the 76ers win, what is the
  (conditional) probability that Iverson played?

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W
.70
I
Wc
.85
.30
W
.40
Ic
.15
Wc
.60
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Joint Probability Table
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Example

• The prevalence rate of a particular disease in
  the population is 1. A blood test for the
  disease is 95 accurate when the patient has the
  disease the test is 99 accurate when the
  patient does not have the disease.
• What is the probability that a patient will test
  positive for the disease.
• Given that a patient tests positive, what is the
  probability that they actually have the disease?

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Example

• D Patient has disease
• DcPatient doesnt have disease
• Patient tests positive for disease
• - Patient tests negative for disease
• Given P(D).01, P(D).95, P(-Dc).99
  

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.95
D
-
.01
.05

.01
Dc
.99
-
.99
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Joint Probability Table
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• Example The following Table describes the 62
  million long-form federal tax returns filed with
  Internal Revenue Services (IRS) in 1996 and the
  percentage of those returns that were audited by
  the IRS.

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• If a tax filer is randomly selected from this
  population of tax filers, what is the probability
  that the tax filer was audited?
• What is the probability that a randomly-selected
  tax filer had an income of below 25,000 and was
  audited?
• If a randomly-selected tax filer was audited,
  what is the probability that the tax filer had an
  income of 50,000-99,999? An income of 100,000
  or more?

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Random Variables (RVs)

• A random variable is a function that assigns a
  numerical value to each simple event in the
  sample space.
• Discrete RV Takes countable number of possible
  values
• Continuous RV Takes uncountable number of
  possible values

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Countable vs. Uncountable

• Countable (finite) e.g., 1, 2, 3
• 0.5,
  1.0, 1.5
• Countable (infinite) e.g., 1, 2, 3,.
• 
  0.5,1.0,1.5,
• Uncountable e.g.,

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Discrete or Continuous RV?

• a) The number of defective items in a batch of 10
  items.
• b) The number of phone calls received by a
  company in 1 hour.
• c) The time between successive phone calls
  received by the company.

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Example

• The number of accidents that occur on a busy
  stretch of highway is a random variable.
• What are the possible values of this random
  variable?
• Are the values countable? Explain.
• Is there a finite number of values?
• Is the random variable discrete or continuous?

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Example

• The distance a car travels on one tank of
  gasoline is a random variable.
• What are the possible values of this random
  variable?
• Are the values countable? Explain.
• Is there a finite number of values?
• Is the random variable discrete or continuous?

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Discrete Probability Distributions

• A discrete probability distribution is a table,
  formula, or graph that lists all possible values
  a discrete random variable can assume, together
  with their associated probabilities.

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Example Two Coin Flips

• Let X denote the total number of heads

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• Probability Table
• Probability formula

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Graph of Probability Distribution
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Requirements of Discrete Probability Distributions

• Probabilities between 0 and 1
• Probabilities sum to 1

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Example

• Determine which of the following are not valid
  probability distributions, and explain why not.

(a)
(b)
(c)
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Example

• Let X represent the number of people in an
  American household. According to the Statistical
  Abstract of the United States 1993, the
  probability distribution of X is as follows
  (rounded to two decimal places).

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• a) What is the probability of a randomly selected
  household having fewer than 4 people?
• b) What is the probability of a randomly selected
  household having between 2 and 5 (inclusive)
  people?
• c) What is the probability of a randomly selected
  household having more than 6 people?
• d) If a household has fewer than 4 people, what
  is the chance that is has exactly 2 people?

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Snickers Example

• A goodie box contains 10 candy bars 8 Milky Ways
  and 2 Snickers. If you reach into the box twice
• What is the probability distribution for the
  number of Snickers bars you obtain?
• What is the chance you get at least one snickers?

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X
S
7/9
S
8/10
M
2/9
S
8/9
M
2/10
M
1/9
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Relationship to Relative Frequencies
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Population Mean

• Recall from Chapter 4 the definition of a
  population mean

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Expected Value

• The expected value or mean of a discrete random
  variable is defined as

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General Formula

• The expected value of any function g(x), can be
  computed as

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Variance and Standard Deviation

• The variance of a discrete RV is defined as
• The standard deviation is given by

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Shortcut Formula for Variance
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Coin Tossing Example

• X number of heads in 3 coin tosses
• Probability distribution

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Coin Tossing Example
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Coin Tossing Example
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Laws of Expected Value
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Laws of Variance
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Graphing Discrete Probability Distributions in
JMP-IN

• Click File/New/Data Table or Ctrl-n
• Create a column with x values.
• Create a column with corresponding p(x) values.
• Click Graph/Overlay Plot
• Highlight x variable and click X
• Highlight p(x) variable and click Y
• Click OK
• Click Overlay/Y Options/Needle
• Right-click on the y-axis and click Axis
  Settings
• Set Minimum0.

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Example

• A company is interested in hiring a person with
  an MBA degree and at least 2 years experience in
  a marketing department of a computer products
  firm. The company's personnel department has
  determined that it will cost the company 1,000
  per job candidate to collect the required
  background information and to interview the
  candidate. As a result, the company will hire the
  first qualified person it finds and will
  interview no more than three candidates. The
  company has received job applications from four
  persons who appear to be qualified but, unknown
  to the company, only one actually possesses the
  required background. Candidates to be interviewed
  will be randomly selected from the pool of four
  applicants.

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• a) Construct the probability distribution for the
  total cost to the firm of the interviewing
  strategy.
• b) What is the probability that the firm's
  interviewing strategy will result in none of the
  four applicants being hired?
• c) What is the expected total cost of the
  interviewing strategy?

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• P(none qualified)
• E(X)

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Example

• Second-year business students at a certain
  university are required to take 10 one-semester
  courses. Suppose that the number of courses in
  which a student will receive a grade of A has a
  discrete uniform distribution (that is, each
  possible number has the same probability of
  occurrence).

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• What are the possible values of the random
  variables and their probabilities?
• What is the probability that a second-year
  business student receives an A in exactly three
  courses?
• What is the probability that a second-year
  business student receives an A in more than 10
  courses?
• What is the probability that a second-year
  business students highest grade is lower than an
  A?

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Example 2

• The number of training units that must be passed
  from a complex computer software program is
  mastered varies from one to five, depending on
  the student. After much experience , the
  software manufacturer has determined the
  probability distribution that describes the
  fraction of users mastering the software after
  each number of training units

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• Calculate the mean and standard deviation of the
  number of training units to master the program.
• If the firm wants to ensure that at least 75 of
  the students master the program, what is the
  minimum number of training units that must be
  administered? At least 90?

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