Ch 6 Discrete Probability Distributions

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Ch 6 Discrete Probability Distributions

• Goals
• Define Probability Distribution and random
  variable.
• Distinguish Discrete and Continuous probability
  distributions
• Calculate mean, variance and standard deviation
  of discrete probability diostributions.
• Binomial, Hypergeometric, and Poisson
  distributions

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Ch 6 Discrete Probability Distributions

• Defn A random variable is a numeric value
  determined by the outcome of an experiment.
• Defn A probability distribution is a listing of
  all possible outcomes of an experiment and the
  probability associated with each outcome.

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Ch 6 Discrete Probability Distributions

• Properties of Probability Density function, p(X)
  where X is a random variable
• The sum of all probabilities associated with each
  outcome of an experiment is 1
• Sp 1.
• The probability associated with an outcome is
  between 0 and 1.
• 0lt p lt 1
• Footnote S sum of

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Ch 6 Discrete Probability Distributions

• Mean SxP(x)
• Variance S(x-µ)2P(x) Sx2 P(x) µ2
• Discrete random variables have gaps between
  each data.
• Continuous random variables is gap-less (ref
  Dedekin cuts and Cantor sets)
• Cumulative probability distributions(Excel)
• Footnote S sum of µ population mean

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Ch 6 Discrete Probability Distributions
Commonly-encountered Distributions

• Discrete distributions well consider 3
• Binomial
• Hypergeometric
• Poisson
• Continuous distribution well consider 1
• Normal

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Ch 6 Discrete Probability Distributions

• Binomial Distribution
• 2 mutually exclusive categories
• Probability of a success remains same
• Trials are independent
• Count the number of successes in a fixed number
  of trials
• P(x) nCxpx(1-p)n-x
• Mean np Variance np(1-p)
• Recall nCx n!/(x!n-x!)
• Key descriptive words repetition, with
  replacement, bi-names
• Footnote p probability of success in a given
  trial

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Binomial Probability Distribution
6-19

• To construct a binomial distribution, let
• n be the number of trials
• x be the number of observed successes
• be the probability of success on each trial
• .. Lets walk-through a binomial experiment

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.. Lets walk-through a binomial experiment
6-19

• Say roll a die 4 times and look for a 3.
• Here n 4 trials where x 1 success(a 3) was
  observed and 1/6 .
• What is the probability of this occurring (if we
  are not concerned about when in the four rolls it
  occurs)?_____(1/6)1(5/6)3_________________
• How many orders can the 3 turn up in 4 rolls of
  a die?_____C(4,1)_____
• So, what is the probability of a 3 in 4 trials?
• ___c(4,1) (1/6)1(5/6)3 ___
• What if you wanted to know the probability of 2
  3s?
• Recall nCx n!/(x!n-x!)

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Binomial Probability Distribution
6-20

• So, the formula for the binomial probability
  distribution is
• Recall nCx n!/(x!n-x!)

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Ch 6 Discrete Probability Distributions

• Hypergeometric Distribution
• Probability of success not the same
• Count the number of successes in a fixed number
  of trials
• P(x) SCxN-SCn-x/NCn
• Key descriptive words repetition, without
  replacement 2-colored balls in urn model

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Ch 6 Discrete Probability Distributions

• Generalizing the Hypergeometric Distribution
• Probability of success not the same
• Count the number of successes of multiple events
  in a fixed number of trials
• P(x) SCx RCrN-S-RCn-x-r/NCn
• Key descriptive words repetition, without
  replacement 3-colored balls in urn model
• Recall nCx n!/(x!n-x!)

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Ch 6 Discrete Probability Distributions

• Poisson distribution
• Limiting form of the binomial when n is large or
  unbounded and p is small.
• Assumptions same as binomial.
• P(x) (µxe-µ)/x!
• e 2.71828
• µ Mean np u. avg. successes in an
  interval
• Mean np Variance
• Key descriptive words repetition, with
  replacement aka. Law of Improbable Events
• Recall x! x-factorial (eg. 4!4321)