Statistics

1
Statistics Data Analysis

• Course Number B01.1305
• Course Section 31
• Meeting Time Wednesday 6-850 pm

CLASS 3
2
Class 3 Outline

• Brief review of last class Probability Trees
• Questions on homework
• Chapter 4 Probability Distributions
• Chapter 5 Some Special Distributions

3
Review of Last Class

• Introduction to probability
• Ways of determining probabilities
• Rules for combining probabilities
• Conditional probabilities
• Probability Trees

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Combining Events

• The union A ? B is the event consisting of all
  outcomes in A or in B or in both.
• The intersection A ? B is the event consisting of
  all outcomes in both A and B.
• If A ? B contains no outcomes then A, B are said
  to be mutually exclusive .
• The Complement of the event A consists of
  all outcomes in the sample space S which are not
  in A.

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Conditional Probability (cont.)
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Statistical Independence

• Events A and B are statistically independent if
  and only if P(BA) P(B). Otherwise, they are
  dependent.
• If events A and B are independent, then P(A ? B)
  P(A)P(B)

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Probability Trees
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Constructing Probability Trees

• Events forming the first set of branches must
  have known marginal probabilities, must be
  mutually exclusive, and should exhaust all
  possibilities
• Events forming the second set of branches must be
  entered at the tip of each of the sets of first
  branches. Conditional probabilities, given the
  relevant first branch, must be entered, unless
  assumed independence allows the use of
  unconditional probabilities
• Branches must always be mutually exclusive and
  exhaustive

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Lets Make a Deal

• In the show Lets Make a Deal, a prize is hidden
  behind on of three doors. The contestant picks
  one of the doors.
• Before opening it, one of the other two doors is
  opened and it is shown that the prize isnt
  behind that door.
• The contestant is offered the chance to switch to
  the remaining door.
• Should the contestant switch?
• Solve by making a tree

10
Employee Drug Testing

• A firm has a mandatory, random drug testing
  policy
• The testing procedure is not perfect.
• If an employee uses drugs, the test will be
  positive with probability 0.90.
• If an employee does not use drugs, the test will
  be negative 95 of the time.
• Confidential sources say that 8 of the employees
  are drug users
• 8 is an unconditional probability 90 and 95
  are conditional probabilities

11
Employee Drug Testing (cont.)

• Create a probability tree and verify the
  following probabilities
• Probability of randomly selecting a drug user who
  tests positive 0.072
• Probability of randomly selecting a non-user who
  tests positive 0.046
• Probability of randomly selecting someone who
  tests positive 0.118
• Conditional probability of testing positive given
  a non-drug user 0.05

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Statistical Modeling

• Statistical modeling is the process of creating
  mathematical representations that reflect
  physical phenomena and make use of any available
  data or information
• Possible uses
• Description
• Prediction
• Optimization
• uncertainty analysis
• Statistical models may integrate real-world data
  with results from physical experiments and
  computer codes.

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CHAPTER 4
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Chapter Goals

• To understand the concepts of probability
  distributions
• To be able to calculate the expected value and
  standard deviation of a distribution
• To understand the difference between sample mean
  and expected value

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Random Variables

• A quantity that takes on different values
  depending on chance
• Next quarters sales for a given company
• The proportion of interviewees that express and
  intention to buy
• Your day- trading profits for next year
• The number of free throws out of 10 a player
  makes
• Number of defective products produced next week
• A random variable is the result of a random
  experiment in the abstract sense, before the
  experiment is performed
• The value the random variable actually assumes is
  called an observation
• A probability distribution is the pattern of
  probabilities a random variable assumes

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Random Variables (cont)

• You can think of your data set as observations of
  a random variable resulting from several
  repetitions of a random experiment
• We associate the random variable with a
  population and view observations of the random
  variable as data
• Example
• Suppose we toss a coin five times. The observed
  data set is a sequence of zeros and ones, such as
  1 1 0 1 0. Each of the five digits in this
  sequence represents the outcome of the random
  experiment of tossing a coin once, where 1
  denotes Heads and 0 denotes Tails. We have five
  repetitions of the experiment.

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Types of Random Variables
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Discrete Probability Distribution

• A list of the possible values of a discrete
  random variable, together with their associated
  probabilities
• The probability distribution tells us everything
  we can know about a random variable, before it
  becomes an observation
• Example Distribution of Heads in Two Tosses
• S HH, HT, TH, TT

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Example Discrete Distribution

• When Quality Control testing entails destroying
  the tested product, for obvious economic reasons,
  a sample of items are tested.
• A plant that produces cell phones in equal
  quantities on two production lines.
• Quality control experts determined the plant
  should test three randomly selected phones 2
  from one line and 1 from the other, where the
  number from each line is chosen by flipping a
  coin three times
• Construct the probability distribution of the
  number of phones chosen from Line 1
• Construct sample space
• Probability
• Random variable value

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Example Discrete Distribution

• We want to conduct two one-on-one interviews with
  neurologists to get their opinions on an existing
  drug
• Suppose that a random sample of two neuros is to
  be selected from all neuros consisting of 70 who
  have ever prescribed the drug and 30 who have
  not
• Questions
• List all possible outcomes of the selection
• Assign probabilities
• Define the quantitative variable Y as the number
  of neuros who have prescribed the drug in the
  sample. Specify the possible values that the
  random variable assume and determine the
  probability of each

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Distribution Notation
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Cumulative Distribution Function
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Example CDF

• Suppose a financial firm plans to release a new
  fund. The fund manager has assessed the
  following subjective probabilities for the
  first-year return for a 10,000 investment
• Find the following probabilities, as assessed by
  the fund manager

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Expected Value of Discrete Distribution
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Example Expected Value

• A firm is considering two possible investments.
  The firm assigns rough probabilities of losing
  20 per dollar invested, losing 10, breaking
  even, gaining 10, and gaining 20.
• Let Y be the return per dollar invested in the
  first project and Z the return per dollar
  invested in the second.

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Standard Deviation

• Measure of probability dispersion, variability,
  or risk of a random variable

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Example Standard Deviation

• A firm is considering two possible investments.
  The firm assigns rough probabilities of losing
  20 per dollar invested, losing 10, breaking
  even, gaining 10, and gaining 20.
• Let Y be the return per dollar invested in the
  first project and Z the return per dollar
  invested in the second.

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Standard Deviation (cont)

• Note s2 and s defined above are theoretical
  variance and standard deviation of X. You dont
  need any data to compute them. You just need to
  know the distribution of X.
• The mean of a random variable is NOT the same
  thing as a sample mean. The variance of a random
  variable is NOT the same thing as a sample
  variance.

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Empirical Rule for Random Variables
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Continuous Random Variables

• As with discrete probability distribution
  functions, one can also determine the expected
  value and standard deviation of continue PDFs
• Mathematical definitions for continuous random
  variables necessarily involve calculus
• Sums are simply replaced by integrals
• This is beyond the expectations for this class

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Example

• A call option on a stock is being evaluated.
• If the stock goes down, the option expires and is
  worthless. If it goes up, the payoff depends on
  how high the stock goes.
• Assume a discrete payoff distribution
• Questions
• What is the expected value of the payoff?
• What is the standard deviation of the payoff?
• Find the probability that the option will pay at
  least 15
• Find the probability that the option will pay
  less than 20

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Chapter 5

• Some Special Probability Distributions

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Chapter Goals

• Introduce some special, often used distributions
• Understand methods for counting the number of
  sequences
• Understand situations consisting of a specified
  number of distinct success/failure trials
• Understanding random variables that follow a
  bell-shaped distribution

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Counting Possible Outcomes

• In order to calculate probabilities, we often
  need to count how many different ways there are
  to do some activity
• For example, how many different outcomes are
  there from tossing a coin three times?
• To help us to count accurately, we need to learn
  some counting rules
• Multiplication Rule If there are m ways of
  doing one thing and n ways of doing another
  thing, there are m times n ways of doing both

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Example

• An auto dealer wants to advertise that for 20G
  you can buy either a convertible or 4-door car
  with your choice of either wire or solid wheel
  covers.
• How many different arrangements of models and
  wheel covers can the dealer offer?

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Counting Rules

• Recall the classical interpretation of
  probabilityP(event) number of outcomes
  favoring event / total number of outcomes
• Need methods for counting possible outcomes
  without the labor of listing entire sample space
• Counting methods arise as answers to
• How many sequences of k symbols can be formed
  from a set of r distinct symbols using each
  symbol no more than once?
• How many subsets of k symbols can be formed from
  a set of r distinct symbols using each symbol no
  more than once?
• Difference between a sequence and a subset is
  that order matters for a sequence, but not for a
  subset

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Counting Rules (cont)

• Create all k3 letter subsets and sequences of
  the r5 letters A, B, C, D and E
• How many sequences are there?
• How many subsets are there?

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Example

• A group of three electronic parts is to be
  assembled into a plug-in unit for a TV set
• The parts can be assembled in any order
• How many different ways can they be assembled?
• There are eight machines but only three spaces on
  the machine shop floor.
• How many different ways can eight machines be
  arranged in the three available spaces?
• The paint department needs to assign color codes
  for 42 different parts. Three colors are to be
  used for each part. How many colors, taken three
  at a time would be adequate to color-code the 42
  parts?

39
Review Sequence and Subset

• For a sequence, the order of the objects for each
  possible outcome is different
• For a subset, order of the objects is not
  important

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Counting Rules (cont)
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Binomial Distribution

• Percentages play a major role in business
• When percentage is determined by counting the
  number of times something happens out of the
  total possibilities, the occurrences might
  following a binomial distribution
• Examples
• Number of defective products out of 10 items
• Of 100 people interviewed, number who expressed
  intention to buy
• Number of female employees in a group of 75
  people
• Number of Independent Party votes cast in the
  next election

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Binomial Distribution (cont)

• Each time the random experiment is run, either
  the event happens or it doesnt
• The random variable X, defined as the number of
  occurrences of a particular event out of n trials
  has a binomial distribution if
• For each of the n trials, the event always has
  the same probability ? of happening
• The trials are independent of one another

43
Example Binomial Distribution

• You are interested in the next n3 calls to a
  catalog order desk and know from experience that
  60 of calls will result in an order
• What can we say about the number of calls that
  will result in an order?
• Create a probability tree
• Questions
• What is the expected number of calls resulting in
  an order
• What is the standard deviation

44
Binomial Distribution the Easy Way
Number of Occurrences, X Proportion or Percentage
Mean E(X) n ? E(p) ?
Standard Deviation ?X(n ?(1- ?))0.5 ?p(?(1- ?)/n)0.5
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Finding Binomial Probabilities
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Example Binomial Probabilities

• How many of your n6 major customers will call
  tomorrow?
• There is a 25 chance that each will call
• Questions
• How many do you expect to call?
• What is the standard deviation?
• What is the probability that exactly 2 call?
• What is the probability that more than 4 call?

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Example

• Its been a terrible day for the capital markets
  with losers beating winners 4 to 1
• You are evaluating a mutual fund comprised of 15
  randomly selected stocks and will assume a
  binomial distribution for the number of
  securities that lost value
• Questions
• What assumptions are being made?
• How many securities do you expect to lose value?
• Find the probability that 8 securities lose value
• What is the probability that 12 or more lose
  value?

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Computing Tutorial

• Simulation
• Calculating probabilities

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Homework 3

• To be handed out in class

50
Next Time

• Normal distribution
• Statistical Inference