Chapter 7 Probability Distributions, Information about the Future

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Chapter 7Probability Distributions,
Information about the Future
.
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• Random Processes

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Random Variable

• A random variable is the numerical outcome of a
  random (non-deterministic) process.
• Intuitively, any numerically measured variable
  that possesses an uncertain outcome is a random
  variable.

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Probability Distribution

• A probability distribution is a model which
  describes a specific kind of random process.
• Specifically, a probability distribution connects
  a probability to each value the random variable
  can assume.

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Probability Models

• Probability models are excellent descriptors of
  random processes.
• The following is a probability distribution (a
  model) for the outcome of a coin toss.

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• Types of Random Variables

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Quantitative Random Variables

• Quantitative random variables are divided into
  two classes.
• ) Discrete
• ) Continuous

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Discrete Random Variables

• A discrete random variable is a random variable
  which has a countable number of possible
  outcomes.
• The values that many discrete random variables
  assume are the counting numbers from 0 to N,
  where N depends upon the nature of the variable.
• Example The number of pages in a standard math
  textbook is a discrete random variable.

Statistics
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Continuous Random Variables

• A continuous random variable is a random variable
  that can assume any value on a continuous
  segment(s) of the real number line.
• Heights, weights, volumes, and time measurements
  are usually measured on a continuous scale.
• These measurements can take on any value in some
  interval.

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Discrete or Continuous?

• Classify the following as either a discrete
  random variable or a continuous random variable.
• 1. the speed of a train
• 2. the possible scores on the SAT exam
• 3. the number of pizzas eaten on a college
  campus each day
• 4. the daily takeoffs at Chicagos OHare
  Airport
• 5. the highest temperatures in Maine and Florida
  tomorrow

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Answers

• 1. the speed of a train
• continuous random variable
• 2. the possible scores on the SAT exam
• discrete random variable
• 3. the number of pizzas eaten on a college
  campus each day
• discrete random variable
• 4. the daily takeoffs at Chicagos OHare
  Airport
• discrete random variable
• 5. the highest temperatures in Maine and Florida
  tomorrow
• continuous random variable

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Naming Convention

• Capital letters, such as X, will be used to refer
  to the random variable.
• example
• X number of cows in Texas
• Small letters, such as x, will refer to a
  specific value of the random variable.
• example
• x 1,498,000 cows in Texas
• Often the specific values will be subscripted x1,
  x2, ..., xn.

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Describing a Discrete Random Variable

• State (Describe) the variable.
• List all of the possible values of the variable.
• Determine the probabilities of these values.

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Example 1 (die tossing)

• Random Phenomenon Toss a die and observe the
  outcome of the toss.
• X ?
• What are the possible values of X?
• What are the probabilities of each value?

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Example 1 - Solution

• Identify the Random Variable X
  outcome of toss of die
• All possible Values Integers between 1 and 6.
  In this instance x1 1, x2 2, ..., x6 6.
• Probability Distribution The outcomes of the
  toss of a die and their probabilities are given
  in the table. The probabilities are
  deduced using the classical method and the
  assumption of a fair die.

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Example 2

• Random Phenomenon The head nurse of the
  pediatric division of the Sisters of Mercy
  Hospital is trying to determine the capacity
  requirement for the nursery. She realizes that
  the number of babies born at the hospital each
  day is a random variable. And, she will have to
  develop a description of the randomness in order
  to develop her plan.
• X ?
• What are the possible values of X?
• What are the probabilities of each value?

• Not all discrete random variables have easily
  definable probability distributions.

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Example 2 - Solution

• Identify the Random Variable X
  of babies born at hospital each day
• Range of All possible Values Integers between 0
  and some large positive number.
• Probability Distribution Unknown, but could be
  estimated using the relative frequency idea in
  conjunction with historical data on hospital
  births.

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• Discrete Probability Distributions

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Discrete Probability Distributions

• The random variable concept is so general, that
  it is not very useful by itself.
• What would be useful is to determine what
  numerical values the random variable could assume
  and assess the probabilities of each of these
  values.

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Discrete Probability Distributions

• A discrete probability distribution consists of
  (a list of) all possible values of the random
  variable with their associated probabilities.
• The association of the possible values of a
  random variable with their respective
  probabilities can be expressed in three different
  forms in a table, in a graph, and in an
  equation.

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Characteristics

• Discrete probability distributions always have
  two characteristics
• 1. The sum of all of the probabilities must
  equal 1.
• The probability of any value must be between 0
  and 1, inclusively.
• Relative frequencies also share these properties

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Example 3 (Daily Sales)

• K. J. Johnson is a computer salesperson. During
  the last year he has kept records on his computer
  sales. He recognizes that his daily sales
  constitute a random process and wishes to
  determine the probability distribution for daily
  sales.
• The random variable is X number of computers
  sold each day.

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Example 3 - Solution

• The probabilities for this random variable are
  computed in the table based upon 200 days of
  sales data obtained from Mr. Johnsons records
  using the relative frequency concept.

The probability that Mr. Johnson will sell at
least 2 computers each day is calculated as
follows P(X 2) P(X2) P(X3) P(X4)
.3 .2 .2 .7. The probability that Mr.
Johnson will sell at most 2 computers each day is
calculated as follows P(X 2) P(X0)
P(X1) P(X2) .2 .1 .3 .6.
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Example 4, Is this a prob. Distn?

• Tell whether or not the following distribution is
  a probability distribution.
• If the distribution is not a probability
  distribution, give the characteristic which is
  not satisfied by the distribution.

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Example 4 - Solution

• Yes. All probabilities are between 0 and 1, and
  the sum of the probabilities is 1.

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Example 5 , Is this a prob. Distn?

• Tell whether or not the following distribution is
  a probability distribution.
• If the distribution is not a probability
  distribution, give the characteristic which is
  not satisfied by the distribution.

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Example 5 - Solution

• No. The sum of the probabilities is greater than
  one.

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Example 6, Is this a prob. Distn?

• Tell whether or not the following distribution is
  a probability distribution.
• If the distribution is not a probability
  distribution, give the characteristic which is
  not satisfied by the distribution.

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Example 6 - Solution

• No. You can't have negative probabilities.

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Example 7, Is this a prob. Distn?

• Tell whether or not the following distribution is
  a probability distribution.
• P(Xx) , for x 1, 2, 3, 4, 5
• If the distribution is not a probability
  distribution, give the characteristic which is
  not satisfied by the distribution.

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Example 7 - Solution

• No. See table. The sum of the probabilities is
  15/16 which is less than one.
• P(X)x/16 for x1 to 5 only is NOT a probability
  distribution.

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• Expected Value E(X) of a random variable X

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Importance of E(X)

• One of the most important concepts in the
  analysis of random phenomena is the notion of
  expected value.
• Expected value is important because it is a
  summary statistic for a probability distribution.
  
• It can also be used as a criteria for comparing
  alternative decisions in the presence of
  uncertainty.

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What is Expected Value?

• Conceptually, expected value is closely allied
  with the notion of mean or average.
• The expected value is a weighted average, in
  which each possible value of the random variable
  is weighted by its probability.
• Definition
• The expected value of a random variable X is the
  mean of the random variable X. It is denoted by
  E(X) and is given by computing the following
  expression
• E(X) ? x P(Xx)
• ? x P(x)

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Digression on Weighted Averages

• Weighted average of any measurement (say prices
  Pt) is always (?t wtPt )/( ?t wt)
• weighted averages are ubiquitous. Dow Jones
  Industrial average is a weighted average
• see
• http//www.indexarb.com/indexComponentWtsDJ.html
• SP 500 index is similar with weights available
  at
• http//www.indexarb.com/indexComponentWtsSP500.htm
  l

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Average Value

• The expected value of a random variable should be
  very close to the average value of a large number
  of observations from the random process.
• The larger the number of observations collected
  the more likely the expected value will be close
  to the average of the observations.
• For discrete random variables the expected value
  is rarely one of the possible outcomes of the
  random variable.

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E(X) for Daily Sales

• The expected value of the probability
  distribution given in Example 3 (daily
  Sales) is computed in the table.
  
• In the long run, data coming from a random
  process with this distribution should average
  about 2.1.

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Using Expected Values to Compare Alternatives

• Two Investment Opportunities
• By calculating the expected values of the two
  alternatives the information in each distribution
  is condensed to a single point.
• This point characterizes the center of the
  random process and facilitates comparison.
• In the long run, option B would be 500 more
  profitable.
• But on any one investment in option B, you may
  lose as much as 3000 or make as much as 4000.

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Symbols

• The expected value, E(X), is the center point
  for the random process.
• The symbol mx is often used to represent E(X).
• mx E(X)

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• Variance and Standard Deviation of a Discrete
  Random Variable

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Variance of a Discrete Random Variable

• The expected value of a distribution measures
  only one dimension of the random variable (its
  central value).
• To gauge the variability of a random variable we
  need another measure similar to the variance
  measure previously constructed but one which
  accounts for the difference in probabilities of
  the variable.
• The variance of a discrete random variable X is
  given by
• The larger the variance the more variability in
  the outcomes.

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Standard Deviation as a measure of risk

• The standard deviation is computed by taking the
  square root of the variance.
• In the Investment Opportunity problem the
  variance and standard deviation are as follows.
• Option A
• V(X) 3,090,000
• 1,757.84
• Option B
• V(X) 6,640,000
• 2,576.82
• The larger deviation reflects greater variability
  in profits and increased risk.

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Sharpe Ratio

• Risk adjusted returns are compared by computing
  the ratio
• Average return / std. Dev of returns
• Option A 900/1,757.84
• 0.5199199
• Option B 1400/ 2,576.82
• 0.5433053
• Clearly Option B is slightly superior.

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Example 9

• Find the expected value, the variance, and the
  standard deviation for a random variable with the
  following probability distribution.

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Example 9 - Solution
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• Probability Distributions and their Functions

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Where do probability distributions come from?

• In previous examples the distribution is already
  given.
• In the real world there will be very few
  instances in which the probability distribution
  will be conveniently available.
• Probabilities will have to be determined using
  (i) classical, (ii) relative frequency, or (iii)
  subjective probability.
• Probability distributions can be constructed from
  relative frequency distributions( depicted in
  histograms)

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Probability Distribution Functions (p.d.f.)

• Four well known discrete distributions are the
  discrete uniform, binomial, Poisson, and
  hypergeometric.
• Each of the discrete distributions possesses a
  probability distribution function.
• These math functions assign probabilities to each
  value of the random variable.

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Discrete Probability Distribution Function

• Example of discrete p.d.f.
• P(Xx)1/4, if x1,2,3,4
• P(Xx)0, otherwise
• This pdf does assign some value to each possible
  discrete number which can be the value of X.
• All probability values need not be positive. They
  can be zero!

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Determining Probabilities for a Specific Value
(just plug into the formula)

• To determine the probability for a specific
  value, use the value as the argument to the
  function. Pdf is P(Xx)x2/30. sum is unity
• To determine the probability that X 3,
• P(X3) .
• To determine the probability that X 4,
• P(X4) .

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• The Discrete Uniform Distribution

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Definition

• In the discrete uniform distribution each value
  of the random variable is assigned identical
  probabilities.
• This distribution is one of the simplest
  probability distributions.
• There are many situations in which the discrete
  uniform distribution arises.

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Example 10

• The outcome of the throw of a single die.
• If the die is fair, then each of the outcomes
  is equally likely.
• Resulting probability distribution

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• The Binomial Distribution

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Definition

• A binomial experiment is a random experiment
  which satisfies all of the following conditions
• There are only two outcomes on each trial of the
  experiment.
• One of the outcomes is usually referred to as a
  success, and the other as a failure.
• The experiment consists of n identical trials as
  described in (1).

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Definition Continued

• The probability of success on any one trial is
  denoted by p and does not change from trial to
  trial.
• Note that the probability of a failure is 1- p
  and also does not change from trial to trial.
• The trials are independent.
• The binomial random variable is the count of the
  number of successes in n trials.

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Example 11 (r.v. is of heads in 4 tosses)

• Toss a coin 4 times and record the number of
  heads as the random variable.
• The number of heads in 4 tosses is a binomial
  random variable.

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Pascal Triangle to compute nCx Binomial
coefficients

• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1
• 1 6 15 20 15 6 1
• 4 coin x0,1,2,3,4 and corresp. P(x)
  1/16,4/16,6/16,4/16, 1/16
• Note each row is created from previous row by
  always starting and ending with ones, computing
  sums of numbers from the previous row

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Ex. 11 Continued (r.v. is of heads in 4 tosses)

• There are only 2 outcomes, heads or not heads.
• The experiment will consist of 4 tosses of a
  coin.
• The probability of a getting a head (success) is
  .5 and does not change from trial to trial.
• The outcome of one toss will not affect other
  tosses.
• The variable of interest is the number of heads
  in 4 tosses.

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The Binomial Probability Distribution Function

• where represents the number of possible
  combinations of n objects taken x at a time
  (without replacement) and is given by
• , and 0!
  1
• n the number of trials, and
• p the probability of a success.

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Calculating a Binomial Probability by plugging in
the Binomial formula

• The parameters of the distribution (n and p) as
  well as the value of the random variable must be
  specified.
• For example, to determine the probability that
  x3, given that n4 and p0.5, substitute those
  values in the probability distribution function
  as follows
• Since,

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Example 12

• Calculate for the following combinations of
  x and n.
• A. n 4 and x 2
• B. n 12 and x 8

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Example 12 - Solution

• A.
• B.

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Binomial Tables

• In order to avoid tedious calculations, binomial
  tables containing a large collection of binomial
  distributions have been constructed.
• These tables are found in the Appendix (pp. 523
  -527).

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Example 13

• The random variable X is a binomial random
  variable with n 12 and p .8.
• Using the tables, find the following
• A. the probability that X is at most 4
• B. the probability that X is at least 1
• C. the probability that X is more than 10

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Example 13 - Solution

• A.
• P(X 4)
• P(X0)P(X1)P(X2)P(X3)P(X4)
• .0000.0000.0000.0001.0005
• .0006
• B.
• P(X 1)1-P(X0)1-.00001
• C.
• P(Xgt10)P(X11)P(X12)
• .2062.0687.2749

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The Shape of a Binomial

• If p is small, the distribution tends to be
  skewed with a tail on the right.

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The Shape of a Binomial

• If p is near .5, the distribution is symmetrical.

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The Shape of a Binomial

• If p is large, the distribution tends to be
  skewed with a tail on the left.

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The Expected Value and Variance of a Binomial
Random Variable

• The expected value of a binomial random variable
  can be computed using the simple analytic
  expression
• E(X) np.
• The variance of a binomial random variable can be
  computed using the analytic expression
• V(X) np(1-p) n p q,
• Where q(1-p) by definition.

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Example 14

• The random variable X is a binomial random
  variable with n12 and p.8.
• A. Find the expected value of X.
• B. Find the variance of X.
• C. Find the standard deviation of X.

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Example 14 - Solution

• A. m E(X) np (12)(.8) 9.6
• B. V(X) np(1-p) (12)(.8)(1-.8)
• 1.92
• C. s 1.386

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• The Poisson Distribution

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Poisson vs. Binomial

• The Poisson distribution is similar to the
  binomial in that the random variable represents a
  count of the total number of successes.
• The major difference between the two
  distributions is that the Poisson does not have a
  fixed number of trials.
• Instead, the Poisson uses a fixed interval of
  time or space in which the number of successes
  are recorded.

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Definition

• In order to qualify as a Poisson random variable
  an experiment must meet two conditions
• Successes occur one at a time. That is, two or
  more successes cannot occur at exactly the same
  point in time or exactly at the same point in
  space.
• The occurrence of a success in any interval is
  independent of the occurrence of a success in
  any other interval.

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The Poisson Probability Distribution Function

• where the transcendental constant e is the limit
  of (11/n)n as n becomes large without bound
• e 2.71828..., and
• l average number of successes
• Note The variance of the Poisson distribution
  is equal to the mean (l).

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The Shape of the Poisson Distribution
l .3

• As l increases, the shape of the Poisson
  distribution begins to resemble a bell shaped
  distribution.

l 3
l 12
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Poisson Random Variables for Time

• The majority of Poisson applications are related
  to the number of occurrences of some event in a
  specific duration of time.
• The average number of successes that occur
  within the duration of time will define the one
  and only parameter l of the Poisson random
  variable.

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Example 15 (morning calls)

• The number of calls received by an office on
  Monday morning between 800 AM and 900 AM has a
  Poisson distribution with l equal to 4.0.
• X the number of calls received by an office on
  Monday morning between 800 AM and 900 AM
• l 4.0

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Example 15 - A (No morning calls)

• A. Determine the probability of getting no calls
  between eight and nine in the morning.

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Example 15 - B (exactly 5 morning calls)

• B. Calculate the probability of getting exactly
  five calls between eight and nine in the morning.

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Example 15 - C (E(X) of morning calls)

• C. What will be the expected number of calls
  received by the office during this time period?
  What is the variance?
• Remember that l 4.0, and that l is the mean, or
  average number of successes.
• Also, remember that the variance of a Poisson
  distribution is equal to the mean.
• Thus, the expected number of calls is 4.0, and
  the variance is also 4.0.

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Example 15 - D (Plot of morning calls)

• D. Graph the probability distribution of the
  number of calls using values from the Poisson
  distribution tables in the Appendix (pp. 528-532).

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Poisson Random Variables for Length and Space

• There are a number of Poisson applications that
  measure the number of successes in some area or
  length.
• The average number of successes in the area or
  length will define the l parameter of the Poisson
  random variable.

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Example 16 (carpet weaving errors)

• The number of weaving errors in a twenty foot by
  ten foot roll of carpet has a Poisson
  distribution with l equal to 0.1.
• X the number of weaving errors in a 20x10 foot
  roll of carpet
• l 0.1

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Example 16 - A (carpet weaving errors p.d.f.)

• A. Using the distribution tables , construct the
  probability distribution for the carpet. l 0.1

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Example 16 - B (lt2 carpet weaving errors)

• B. What is the probability of observing less
  than 2 errors in the carpet? l 0.1
• P(Xlt2) P(X0) P(X1)
• .9048 .0905 .9953

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Example 16 - C (gt5 carpet weaving errors)

• C. What is the probability of observing more
  than 5 errors in the carpet? l 0.1
• P(Xgt5) 0

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• The Hypergeometric Distribution

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Hypergeometric vs. Binomial

• Similarities
• Both random variables have only two outcomes on
  each trial of the experiment.
• They both count the number of successes in n
  trials of an experiment.

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Hypergeometric vs. Binomial

• Differences
• The hypergeometric distribution differs from the
  binomial distribution in the lack of independence
  between trials. the probability of success
  will vary between trials for the hypergeometic
  pdf .
• In addition, hypergeometric distributions have
  finite populations in which the TOTAL number of
  successes and failures are known.

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The Hypergeometric Probability Distribution
Function

• A the largest number of succ-esses possible
  in population
• N the size of the total population
• n size of the sample drawn

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Example 17 (distn of memory chips)

• Suppose that a shipment from Matsua Semiconductor
  contains 30 memory chips of which two are bad.
• If a memory board requires 16 chips, what is the
  probability distribution for the number of
  defective chips on the memory board?

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Example 17 - Solution

• The random variable under consideration is given
  as
• X number of defective chips on the memory
  board.
• The three parameters of the distribution are
• A 2 (a success in this case is a defective
  chip).
• N 30, and
• n 16.
• The maximum value of X in this case is 2
  min(2,16).

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Example 17 - Solution

96
The Expected Value and Variance of a
Hypergeometric Random Variable

• The expected value of a hypergeometric random
  variable can be obtained using the expression
• The variance of a hypergeometric random variable
  is

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Example 18, E(X) and V(X) for Hypergeomc

• Compute the expected value and variance for the
  random variable for memorychips defined in
  Example 17.
• A 2, N 30, and n 16
• E(X) 16 ( ) 1.067
• If the experiment were repeated many times, the
  average number of defective chips per board would
  be slightly greater than 1.