Probability Distributions

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Probability Distributions

• MATH 224

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Civil Related Example

• Sludge pumps at a landfill site have a 1/50
  chance of breaking down in a given week (roughly
  one breaks per year)
• What is the probability of a site with 6 pumps
  having two broken in the same week?

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Design Questions

• If 5 pumps are needed to maintain safety of
  groundwater, is the risk low enough with these
  pumps?
• If not, what could you do?

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Pump Probabilities

• Probability of at least one broken pump
• Distribution over of working pumps
• Assumptions?

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Known Distributions

• Computing new distribution by hand, or by
  simulation, is a lot of work
• Many common scenarios arise, and have been
  analyzed so the distribution can be computed and
  worked with more easily

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Binomial Distribution

• Distribution of broken pumps is the same as
• ( heads) out of ( tosses of a coin)
• ( of wins) out of ( plays in gambling)
• ( of speeders) found out of ( cars observed)

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Binomial Conditions I

• We repeat a sequence of trials n times
• 6 pumps are run for a week, 50 coins are flipped
• Each trial has only two possible outcomes

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Binomial Conditions II

• Trials are independent
• Probability of events doesnt change from trial
  to trial

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Binomial Parameters

• The binomial distribution needs two numbers
  defined
• n, the number of trials
• p, the probability of one of the outcomes
• one outcome is usually called success
• other outcome has probability _________

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Computing Probabilities

• Knowing n and p, we can compute directly the
  probability of x successful trials

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Pump Example

• Consider one week at the landfill site
• n 6 pumps
• p 49/50 of each pump working
• from 1/50 chance of failure
• We want to know p(0 working), p(1 working), p(6
  working)
• Domain of x, where x working?

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Pump Example in MATLAB

• Look under binomial in Help menu
• ignore negative binomial something related
  which well cover later
• Look for pdf Probability Distribution
  Function

x 06 of pumps working could be any of
these s N 6 6 pumps p 49/50
probability of 'success' Px binopdf(x, N,
p) bar(x, Px)
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Aside bar vs hist

• 'bar' and 'hist' both produce bar charts what's
  the difference?

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Interpreting the Distribution Graph
Px
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Questions answered by Distribution

• What is the probability of exactly 2 pumps
  failing in the same week?
• What is the probability of 2 or more pumps
  failing in the same week?

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Cumulative Distributions

• Second question is of a very common type
• probability of x or more failures
• Can be answered by
• adding up probabilities for P(x), P(x1),
• OR using
• Mathematically, F(x)

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Cumulative Binomial Distribution in MATLAB
x 06 of pumps working could be any of
these s N 6 6 pumps p 49/50
probability of 'success' Fx binocdf(x, N,
p) bar(x, Fx)
Fx
From before, Px
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Cumulative Graph
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Answering the question

• Probability of 2 or more pumps failing in a given
  week is

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Moving on

• Section 3.2, 3.4, 3.5
• Have clickers?