The Negative Binomial and Uniform Distributions

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The Negative Binomial and Uniform Distributions

• MATH 224

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Negative Binomial Distribution

• Assumptions
• Sequence of independent trials
• Each trial is a success or failure
• Probability of success is constant
• The trials continue until a total of r successes
  have been observed
• Measured the number of failures that precede the
  r-th success
• Success can be misleading term

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Negative Binomial Distribution

• Contrast with binomial (non-negative version)

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Examples
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Distribution of Negative Binomial
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Example the 100-year flood

• A hundred-year flood is one which
• happens every one hundred years
• happens, on average, every 100 years
• happens with probability 1/100 in each year

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Probability of occurrence

• You are building a flood-control system for a
  city
• System is to last 30 years
• Do you need to plan to handle a 1-in-100 year
  flood?
• Do you need to plan for 2?

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In MATLAB
p 1/100 (success 1-in-100 flood) r
1 x 130 for design, x 1( gt 100)
for interest x 1200 Px nbinpdf(x, r,
p) bar(x, Px) Fx nbincdf(x, r, p) bar(x,
Fx)
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Graphs
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Interpretation pdf vs cdf
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Interpretation

• Probability of a single 1-in-100 year flood, in
  100 years, is ___________
• In a 30-year design, what is the probability of
• one such flood?
• two such floods?
• one or more such floods?

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Continued
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Continuous vs Discrete

• Binomial and negative binomial are discrete
  distributions because events are discrete
• of pumps, flips, heads are all integers
• Other events are continuous
• rainfall, height, bending, concentration, time
• Some are almost continuous
• marks, age in years, date of birth

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Use of continuous distributions

• Continuous distributions are often used in
  discrete cases if there are a large number of
  possible events
• Consider probability distribution of heads when
  tossing N coins for
• N 10
• N 100
• N 1000

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Graphical Comparison
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Probability Density Functions

• Discrete distributions have a probability of each
  event
• P(6 working pumps) 0.9
• In continuous distributions
• probability of any exact event value is zero

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Probability Density Functions 2

• Probability is assigned to intervals
• Quick reference the uniform distribution
• Pick number evenly from interval 0, 10
• Probability of 7.6827592875287 is_____
• Probability of picking a number between 0 and 2
  is______________

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Graphically

• We draw the continuous version of a histogram,
  f(x), which is uniform over the interval 0, 10

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Properties of continuous PDF

• Given this probability density f(x),
• Probability of x5 exactly is
• Total probability, over all x values, is
• P(2 lt x lt 4) can be computed using

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Cumulative Distributions

• Values like P(2ltxlt4) can be computed using same
  cumulative strategy we used for discrete
  distributions
• Define uniform CDF, F(x) as

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CDF uses

• For uniform distribution on 0, 10, probability
  that a value between 2 and 4 will come up is
  ___________
• More interesting with more interesting
  distributions (non-uniform)