COMMONLY USED PROBABILITY DISTRIBUTION

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COMMONLY USED PROBABILITY DISTRIBUTION

• CHAPTER 3
• BCT2053

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CONTENT

• 3.1 Binomial Distribution
• 3.2 Poisson Distribution
• 3.3 Normal Distribution
• 3.4 Central Limit Theorem
• 3.5 Normal Approximation to the Binomial
  Distribution
• 3.6 Normal Approximation to the Poisson
  Distribution
• 3.7 Normal Probability Plots

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OBJECTIVES

• At the end of this chapter, you should be able to
• Explain what a Binomial Distribution, identify
  binomial experiments and compute binomial
  probabilities
• Explain what a Poisson Distribution, identify
  Poisson experiments and compute Poisson
  probabilities
• Find the expected value (mean), variance, and
  standard deviation of a binomial experiment and a
  Poisson experiment .
• Identify the properties of the normal
  distribution.
• Find the area under the standard normal
  distribution, given various z values.
• Find probabilities for a normally distributed
  variable by transforming it into a standard
  normal variable.

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OBJECTIVES, Cont

• At the end of this chapter, you should be able to
• Find specific data values for given percentages,
  using the standard normal distribution
• Use the central limit theorem to solve problems
  involving sample means for large samples
• Use the normal approximation to compute
  probabilities for a Binomial variable.
• Use the normal approximation to compute
  probabilities for a Poisson variable.
• Plot and interpret a Normal Probability Plot

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3.1 Binomial Distribution

• A Binomial distribution results from a procedure
  that meets all the following requirements
• The procedure has a fixed number of trials ( the
  same trial is repeated)
• The trials must be independent
• Each trial must have outcomes classified into 2
  relevant categories only (success failure)
• The probability of success remains the same in
  all trials

• Example toss a coin, Baby is born, True/false
  question, product, etc ...

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Binomial Experiment or not ?

• An advertisement for Vantin claims a 77 end of
  treatment clinical success rate for flu
  sufferers. Vantin is given to 15 flu patients who
  are later checked to see if the treatment was a
  success.
• A study showed that 83 of the patients receiving
  liver transplants survived at least 3 years. The
  files of 6 liver recipients were selected at
  random to see if each patients was still alive.
• In a study of frequent fliers (those who made at
  least 3 domestic trips or one foreign trip per
  year), it was found that 67 had an annual income
  over RM35000. 12 frequent fliers are selected at
  random and their income level is determined.

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Notation for the Binomial Distribution
Then, X has the Binomial distribution with
parameters n and p denoted by X Bin (n, p)
which read as X is Binomial distributed with
number of trials n and probability of success p
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Binomial Probability Formula
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Examples

• A fair coin is tossed 10 times. Let X be the
  number of heads that appear. What is the
  distribution of X?
• A lot contains several thousand components. 10
  of the components are defective. 7 components are
  sampled from the lot. Let X represents the number
  of defective components in the sample. What is
  the distribution of X ?

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Solves problems involving linear inequalities

• At least, minimum of, no less than
• At most, maximum of, no more than
• Is greater than, more than
• Is less than, smaller than, fewer than

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Examples

• Find the probability distribution of the random
  variable X if X Bin (10, 0.4).
• Find also P(X 5) and P(X lt 2).
• Then find the mean and variance for X.
• A fair die is rolled 8 times. Find the
  probability that no more than 2 sixes comes up.
  Then find the mean and variance for X.

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Examples

• A survey found that, one out of five Malaysians
  say he or she has visited a doctor in any given
  month. If 10 people are selected at random, find
  the probability that exactly 3 will have visited
  a doctor last month.
• A survey found that 30 of teenage consumers
  receive their spending money from part time jobs.
  If 5 teenagers are selected at random, find the
  probability that at least 3 of them will have
  part time jobs.

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Solve Binomial problems by statistics table

• Use Cumulative Binomials Probabilities Table
• n number of trials
• p probability of success
• k number of successes in n trials X
• It give P (X k) for various values of n and p
• Example n 2 , p 0.3
• Then P (X 1) 0.9100
• Then P (X 1) P (X 1) - P (X 0) 0.9100
  0.4900 0.4200
• Then P (X 1) 1 - P (X lt1) 1 - P (X 0) 1
  0.4900 0.5100
• Then P (X lt 1) P (X 0) 0.4900
• Then P (X gt 1) 1 - P (X 1) 1- 0.9100
  0.0900

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Using symmetry properties to read Binomial tables

• In general,
• P (X k X Bin (n, p)) P (X n - k X
  Bin (n,1 - p))
• P (X k X Bin (n, p)) P (X n - k X
  Bin (n,1 - p))
• P (X k X Bin (n, p)) P (X n - k X
  Bin (n,1 - p))
• Example n 8 , p 0.6
• Then P (X 1) P (X 7 p 0.4) P ( 1 - X
  6 p 0.4)
• 1 0.9915 0.0085
• Then P (X 1) P (X 7 p 0.4)
• P (X 7 p 0.4) - P (X 6 p
  0.4)
• 0.9935 0.9915 0.0020
• Then P (X 1) P (X 7 p 0.4) 0.9935
• Then P (X lt 1) P (X gt 7 p 0.4) P ( 1 - X
  7 p 0.4)
• 1 0.9935 0.0065

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Examples

• Given that n 12 , p 0.25. Then find
• P (X 3)
• P (X 7)
• P (X 5)
• P (X lt 2)
• P (X gt 10)
• Given that n 9 , p 0.7. Then find
• P (X 4)
• P (X 8)
• P (X 3)
• P (X lt 5)
• P (X gt 6)

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Example

• A large industrial firm allows a discount on any
  invoice that is paid within 30 days. Of all
  invoices, 10 receive the discount. In a company
  audit, 12 invoices are sampled at random.
• What is probability that fewer than 4 of 12
  sampled invoices receive the discount?
• Then, what is probability that more than 1 of the
  12 sampled invoices received a discount.

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Example

• A report shows that 5 of Americans are afraid
  being alone in a house at night. If a random
  sample of 20 Americans is selected, find the
  probability that
• There are exactly 5 people in the sample who are
  afraid of being alone at night
• There are at most 3 people in the sample who are
  afraid of being alone at night
• There are at least 4 people in the sample who are
  afraid of being alone at night

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4.4 Poisson Distribution

• The Poisson distribution is a discrete
  probability distribution that applies to
  occurrences of some event over a specified
  interval ( time, volume, area etc..)
• The random variable X is the number of
  occurrences of an event over some interval
• The occurrences must be random
• The occurrences must be independent of each other
• The occurrences must be uniformly distributed
  over the interval being used

• Example of Poisson distribution
• The number of emergency call received by an
  ambulance control in an hour.
• The number of vehicle approaching a bus stop in a
  5 minutes interval.
• 3. The number of flaws in a meter length of
  material

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Poisson Probability Formula

• ?, mean number of occurrences in the given
  interval is known and finite
• Then the variable X is said to be
  
• Poisson distributed with
  mean ?
• X Po (?)

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Example

• A student finds that the average number of
  amoebas in 10 ml of ponds water from a particular
  pond is 4. Assuming that the number of amoebas
  follows a Poisson distribution, find the
  probability that in a 10 ml sample,
• there are exactly 5 amoebas
• there are no amoebas
• there are fewer than three amoebas

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Example

• On average, the school photocopier breaks down 8
  times during the school week (Monday - Friday).
  Assume that the number of breakdowns can be
  modeled by a Poisson distribution.
  Find the probability that it breakdowns,
• 5 times in a given week
• Once on Monday
• 8 times in a fortnight

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Solve Poisson problems by statistics table

• Given that X Po (1.6). Use cumulative Poisson
  probabilities table to find
• P (X 6)
• P (X 5)
• P (X 3)
• P (X lt 1)
• P (X gt 10)
• Find also the smallest integer n such that
  P ( X gt n) lt 0.01

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Example

• A sales firm receives, on the average, three
  calls per hour on its toll-free number. For any
  given hour, find the probability that it will
  receive the following
• At most three calls
• At least three calls
• 5 or more calls

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Example

• The number of accidents occurring in a weak in a
  certain factory follows a Poisson distribution
  with variance 3.2.
• Find the probability that in a given fortnight,
• exactly seven accidents happen.
• More than 5 accidents happen.

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Using the Poisson distribution as an
approximation to the Binomial distribution

• When n is large (n gt 50) and p is small (p lt
  0.1), the Binomial distribution X Bin (n, p)
  can be approximated using a Poisson distribution
  with X Po (?) where mean, ? np lt 5.
• The larger the value of n and the smaller the
  value of p, the better the approximation.

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Example

• Eggs are packed into boxes of 500. On average 0.7
  of the eggs are found to be broken when the
  eggs are unpacked.
• Find the probability that in a box of 500 eggs,
• Exactly three are broken
• At least two are broken

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Example

• If 2 of the people in a room of 200 people are
  left-handed, find the probability that
• exactly five people are left-handed.
• At least two people are left-handed.
• At most seven people are left-handed.