Probability distribution

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Probability distribution

• Dr. Deshi Ye
• College of Computer Science, Zhejiang University
• yedeshi_at_zju.edu.cn

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Outline

• Chebyshevs Theorem
• The Poisson distribution process
• The Geometric Distribution
• The Multinomial distribution
• Simulation

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4.5 Chebyshevs theorem

• EX.
• The number of customers who visit a car
  dealers showroom on a Saturday morning is a
  random variable with mean 18 and standard
  variance 2.5.
• Question What probability can we assert that
  there will be more than 8 but fewer than 28
  customers?

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Chebyshevs theorem

• Theorem 4.1. If a probability distribution has
  mean and standard deviation , the
  probability of getting a value which deviates
  from by at least is at most

Symbolically
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Proof.
Since
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Corollary
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EX

• The number of customers who visit a car dealers
  showroom on a Saturday morning is a random
  variable with mean 18 and standard variance 2.5.
• Question What probability can we assert that
  there will be more than 8 but fewer than 28
  customers?

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Solution of EX

• Let X be the number of customers.

Hence k 4
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4.6 Poisson Distribution

• Poisson distribution often serves as a model for
  counts which do not have a natural upper bound.
• Poisson distribution

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Comparing Poisson and binomial

• It is known that 5 of the books bound at a
  certain bindery have defective bindings. Find the
  probability that 2 of 100 books bound by this
  bindery will have defective bindings using
• A) the formula for the binomial distribution
• B) the poisson approximation to the binomial
  distribution

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Solution

• A) x2, n100, p0.05 hence
• B) x2, for the Poisson
  distribution, we

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Poisson approximation to binomial distribution
EX P129
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Poisson distribution

• Mean and variance of Poisson distribution

Proof.
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4.7 Poisson Processes

• On average 0.3 customer/min at a cafeteria,
• Question then what is the probability that 3
  customers will arrive in 5-minute span?

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Random process

• Random process is a physical process that is
  wholly or in part controlled by some sort of
  chance mechanism.
• Characterize time dependence, certain events do
  or do not take place at regular intervals of time
  or throughout continuous intervals of time.

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• Goal find the probability of x success during a
  time interval of length T.
• Divide T into n equal parts of length ?t
• Tn ?t
• 1) The prob. of a success during a very small
  interval ?t is given by a?t
• 2) The prob. of more than one success during a
  small time interval ?t is negligible.
• 3) The prob. of a success during such a time
  interval does not depend on what happen prior to
  that time.

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By Poisson probability

• When n is large enough the probability of x
  success during the time interval T is given the
  corresponding Poisson distribution with the
  parameter

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Solution of EX

• Solution

Consequently
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4.8 The Geometric Distribution

• Suppose that a sequence of trials we are
  interested in the number of the trials on which
  the first success occurs.
• Geometric distribution

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Mean and variance

• Mean of geometric distribution
• Variance of geometric distribution

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4.9 The multinomial distribution
Expansion of
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4.10 Simulation

• Simulation can be useful tools for finding
  approximate probabilities for situation in which
  the actual probabilities are too difficult to
  calculate.
• Approximation will typically be better the more
  repetitions you perform.

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Activities

• Divide students into groups such that each group
  has 3 students and also have 3 cards.
• 1) 3 cards written their own names.
• 2) Randomly shuffle
• 3) Each one pick cards, count number of matches

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Theoretical analysis

• XYZ, ABC, match X-gtA, Y-gtB, C-gtZ
• ABC ACB BAC BCA CAB CBA
• 3 1 1 0 0 1
• Probability
• 0 1 2 3
• 1/3 ½ 0 1/6

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4-Person Analysis
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Homework

• Problems in Textbook (4.32,4.34,4.44,4.47,4.52,4.5
  5,4.59,4.61)