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.

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3.3 Normal Distribution

• A discrete variable cannot assume all values
  between any two given values of the variables.
• A continuous variable can assume all values
  between any two given values of the variables.
• Examples of continuous variables are the heights
  of adult men, body temperatures of rats, and
  cholesterol levels of adults.
• Many continuous variables, such as the examples
  just mentioned, have distributions that are
  bell-shaped, and these are called approximately
  normally distributed variables.

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Example Histograms for the Distribution
of Heights of Adult Women
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Properties of Normal Distribution

• Also known as the bell curve or the Gaussian
  distribution, named for the German mathematician
  Carl Friedrich Gauss (17771855), who derived its
  equation.
• X is continuous where
• and

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The Normal Probability Curve

• The Curve is bell-shaped
• The mean, median, and mode
• are equal and located at the
• center of the distribution.
• The curve is unimodal (i.e., it has only one
  mode).
• The curve is symmetric about the mean, (its shape
  is the same on both sides of a vertical line
  passing through the center.
• The curve is continuous, (there are no gaps or
  holes)
• For each value of X, there is a corresponding
  value of Y.

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The Normal Probability Curve

• The curve never touches the x axis.
  Theoretically, no matter how far in either
  direction the curve extends, it never meets the x
  axisbut it gets increasingly closer.
• The total area under the normal distribution
  curve is equal to 1.00, or 100.
• A Normal Distribution is a continuous, symmetric,
  bell shaped distribution of a variable.

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Area Under a Normal Distribution Curve

• The area under the part of the normal curve that
  lies
• within 1 standard deviation of the mean is
  approximately 0.68, or 68
• within 2 standard deviations, about 0.95, or 95
• within 3 standard deviations, about 0.997, or
  99.7.

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Other Characteristics

• Finding the probability
• Area under curve

Example Given
, Find the value of a and b if
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Shapes of Normal Distributions
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The Standard Normal Distribution

• The standard normal distribution is a normal
  distribution with a mean of 0 and a standard
  deviation of 1.

TIPS
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Different between 2 curves
Area Under the Normal Distribution Curve
Area Under the Standard Normal Distribution Curve
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Finding Area under the Standard Normal
Distribution
GENERAL PROCEDURE

• STEP 1 Draw a picture.
• STEP 2 Shade the area desired.
• STEP 3 Find the correct figure in the following
  Procedure Table (the figure that is similar to
  the one youve drawn).
• STEP 4 Follow the directions given in the
  appropriate block of the Procedure Table to get
  the desired area.

• EXAMPLE
• P (0 lt Z lt 2.34)
• P (-2.34 lt Z lt 0)
• P (0 lt Z lt 0.156)
• P (-1.738 lt Z lt 0)

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Finding Area under the Standard Normal
Distribution

• EXAMPLE
• P (0.21 lt Z lt 2.34)
• P (-2.134 lt Z lt -0.21)
• P (0.67 lt Z lt 1.156)
• P (-1.738 lt Z lt -0.79)

• EXAMPLE
• P (Z gt1.25)
• P (-2.13lt Z)
• P (Z gt2.099)
• P (-0.087lt Z)

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Finding Area under the Standard Normal
Distribution

• EXAMPLE
• P (Z lt 1.21)
• P (Z lt 2.099)
• P (Z lt 0.512)

• EXAMPLE
• P (-0.21 lt Z lt 2.34)
• P (-2.134 lt Z lt 0.21)
• P (-0.67 lt Z lt 1.156)
• P (Z lt 0.79)

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Finding Area under the Standard Normal
Distribution

• EXAMPLE
• P (Z gt-1.25)
• P (Z gt-2.13)
• P (Z gt-0.087)

• EXAMPLE
• P (Z gt2.34)
• P (Z gt0.147)

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Example

• Given X N(110,144), find
• P (110 lt X lt 128) (d) P (X gt 170)
• P (X lt 150) (e) P (98 lt X lt 128)
• P (X gt 130) (f) P (X lt 60)

Transform the original variable X where to a
standard normal distribution variable Z where
TIPS
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Examples
TIPS

• If Z N(0,1), find the value of a if
• P(Z lt a) 0.9693
• P(Z lt a) 0.3802
• P(Z lt a) 0.7367
• P(Z lt a) 0.0793
• If X N(µ,36) and P ( X gt 82) 0.0478, find µ.
• If X N(100, s ²) and P ( X lt 82) 0.0478, find
  s.

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Applications of the Normal Distribution

• 1. The mean number of hours an American worker
  spends on the computer is 3.1 hours per workday.
  Assume the standard deviation is 0.5 hour. Find
  the percentage of workers who spend less than 3.5
  hours on the computer. Assume the variable is
  normally distributed.
• 2. Length of metal strips produced by a machine
  are normally distributed with mean length of 150
  cm and a standard deviation of 10cm.
• Find the probability that the length of a
  randomly selected is
• a) Shorter than 165 cm
• b) within 5cm of the mean

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Applications of the Normal Distribution

• 3. Time taken by the Milkman to deliver to the
  Jalan Indah is normally distributed with mean of
  12 minutes and standard deviation of 2 minutes.
  He delivers milk everyday. Estimate the numbers
  of days during the year when he takes
• a) longer than 17 minutes
• b) less than ten minutes
• c) between 9 and 13 minutes
• 4. To qualify for a police academy, candidates
  must score in the top 10 on a general abilities
  test. The test has a mean of 200 and a standard
  deviation of 20.
• Find the lowest possible score to qualify.
  Assume the test scores are normally distributed.

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Applications of the Normal Distribution

• 5. The heights of female student at a particular
  college are normally distributed with a mean of
  169cm and a standard deviation of 9 cm.
• a) Given that 80 of these female students have
  a
• height less than h cm. Find the value of
  h.
• b) Given that 60 of these female students have
  a
• height greater than y cm. Find the value
  of y.
• 6. For a medical study, a researcher wishes to
  select people in the middle 60 of the population
  based on blood pressure. If the mean systolic
  blood pressure is 120 and the standard deviation
  is 8, find the upper and lower readings that
  would qualify people to participate in the study.

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3.4 The Central Limit Theorem
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The Central Limit Theorem
If and n sample is
selected, then Use a standard normal
distribution variable Z where
TIPS
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Examples

• 1. A. C. Neilsen reported that children between
  the ages of 2 and 5 watch an average of 25 hours
  of television per week. Assume the variable is
  normally distributed and the standard deviation
  is 3 hours.
• If 20 children between the ages of 2 and 5 are
  randomly selected, find the probability that the
  mean of the number of hours they watch television
  will be greater than 26.3 hours.
• 2. The average age of a vehicle registered in
  the United States is 8 years, or 96 months.
  Assume the standard deviation is 16 months.
• If a random sample of 36 vehicles is selected,
  find the probability that the mean of their age
  is between 90 and 100 months.

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Examples

• 3. The average number of pounds of meat that a
  person consumes a year is 218.4 pounds. Assume
  that the standard deviation is 25 pounds and the
  distribution is approximately normal.
• a. Find the probability that a person selected
  at random
• consumes less than 224 pounds per year.
• b. If a sample of 40 individuals is selected,
  find the
• probability that the mean of the sample will
  be less than
• 224 pounds per year.

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3.5 Normal approximation to the Binomial
Distribution

• If X Bin (n, p) and n and p are such that np gt
  5 and np gt 5 where q 1 p then X N (np, npq)
  approximately.
• The continuity correction is needed when using a
  continuous distribution (normal) as an
  approximation for a discrete distribution
  (binomial), i.e

TIPS class boundary
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Examples

• 1. In a sack of mixed grass seeds, the
  probability that a seed is ryegrass is 0.35. Find
  the probability that in a random sample of 400
  seeds from the sack,
• less than 120 are ryegrass seeds
• between 120 and 150 (inclusive) are ryegrass
• more than 160 are ryegrass seeds
• 2. Find the probability obtaining 4, 5, 6 or 7
  heads when a fair coin is tossed 12 time using a
  normal approximation to the binomial distribution

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3.6 Normal approximation to the Poisson
Distribution

• If X Po (?) and ? gt 15, then X can be
  approximated by Normal distribution with X N
  (?, ?)
• The continuity correction is also needed.

• If X Po (35), use the normal approximation to
  find
• P ( X 33)
• P ( X gt 37)
• P (33 lt X lt 37)
• P ( X 37)

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Examples

• 2. A radioactive disintegration gives counts that
  follow a Poisson distribution with mean count of
  25 per second.
• Find the probability that in one-second interval
  the count is between 23 and 27 inclusive.
• 3. The number of hits on a website follows a
  Poisson distribution with mean 27 hits per hour.
  
• Find the probability that there will be 90 or
  more hits in three hours.

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3.7 Normal Probability Plots

• To determine whether the sample might have come
  from a normal population or not
• The most plausible normal distribution is the one
  whose mean and standard deviation are the same as
  the sample mean and standard deviation

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How to plot?

• Arrange the data sample in ascending (increasing)
  order
• Assign the value (i -0.5) / n to xi
• to reflect the position of xi in the ordered
  sample. There are i - 1 values less than xi ,
  and i values less than or equal to xi . The
  quantity (i -0.5) / n is a compromise between the
  proportions (i - 1) / n and i / n
• Plot xi versus (i -0.5) / n
• If the sample points lie approximately on a
  straight line, so it is plausible that they came
  from a normal population.

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Examples

• A sample of size 5 is drawn. The sample, arranged
  in increasing order, is
• 3.01 3.35 4.79 5.96 7.89
• Do these data appear to come from an
  approximately
• normal distribution?
• The data shown represent the number of movies in
  US for 14-year period.
• 2084 1497 1014 910 899 870 859
• 848 837 826 815 750 737 637
• Do these data appear to come from an
  approximately
• normal distribution?

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Conclusion

• Statistical Inference involves drawing a sample
  from a population and analyzing the sample data
  to learn about the population.
• In many situations, one has an approximate
  knowledge of the probability mass function or
  probability density function of the population.
• In these cases, the probability mass
• or density function can often be well
• approximated by one of several standard
• families of curves or function discussed
• in this chapter.

Thank You