Statistics 270 - Lecture 10

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Statistics 270 - Lecture 10
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• Last day/Today Discrete probability
  distributions
• Assignment 3 Chapter 2 44, 50, 60, 68, 74, 86,
  110

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Common Discrete Probability Distributions

• There are several probability distributions that
  describe a large variety of random phenomena
• Will consider 5 of these
• Discrete Uniform
• Bernoulli
• Binomial
• Hypergeometric
• Poisson

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Discrete Uniform Distribution

• Have seen this already
• Random variable X has k possible outcomes, each
  equally likely
• pmf
• Mean and Variance

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Bernoulli Distribution

• Have seen this already
• Random variable X has 2 possible outcomes
• pmf
• Mean and Variance

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Binomial Distribution

• Count the number of successes in n independent
  Bernoulli trials
• Binomial Experiment
• Have n trials, where n is fixed in advance of the
  experiment
• Each trial results in one of two possible
  outcomes (success or failure)
• The outcomes are independent
• The probability of success is constant

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Binomial Distribution

• Count the number of successes in n independent
  Bernoulli trials
• Binomial Experiment
• Have n trials, where n is fixed in advance of the
  experiment
• Each trial results in one of two possible
  outcomes (success or failure)
• The outcomes are independent
• The probability of success is constant

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Binomial Distribution

• Let X denote the number of successes in n
  independent Bernoulli trials
• Then the rv, X, is said to be a binomial rv
• pmf

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Binomial Distribution

• Mean
• Variance

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Example

• A baseball player has a 300 batting average
• What is the expected number of hits in 25 at
  bats?

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Example

• According to CTV News, the 2006 Federal Election
  results were

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Example

• Ten voters from across the country are randomly
  selected and the number of of Conservative voters
  is counted
• Is this a Binomial experiment?
• What is the probability that 6 of them voted for
  the Conservatives?
• What is the expected number of Conservative votes
  in such a sample?

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Hyper-geometric Distribution

• Count the number of successes in n trials from a
  population with N individuals and M successes
• Assumptions
• Have n trials, where n is fixed in advance of the
  experiment
• Population has N individuals (finite population)
• There are two possible outcomes (success and
  failure) and there are M successes in the
  population
• A sample of n individuals is taken WITHOUT
  replacement

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Hyper-geometric Distribution

• Let X denote the number of successes in a sample
  of size n, without replacement
• Then the rv, X, is said to be a hyper-geometric
  rv
• pmf

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Hyper-geometric Distribution

• Mean
• Variance

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Example (Chapter 3 - 64)

• A digital camera comes in either a 3 or 4
  mega-pixel version
• A store receives 15 cameras and 6 are the 3
  mega-pixel version
• Suppose 5 of these are randomly selected and
  stored behind the counter
• Let X denote the number of 3 mega-pixel cameras
  stored behind the counter
• Compute P(X2) and P(X lt2)