Geometric Distributions

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Geometric Distributions

• Mr. Luitwieler
• AP Statistics
• Holland Hall School

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What is a Geometric Distribution?

• In laymans terms
• How many outcomes must occur to get FIRST
  success?
• It is the PROBABILITY distribution of these
  different observations.
• Example Flip a coin until I get a HEADS.
• Example Throw a dart until I get a BULLSEYE.

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GEOMETRIC vs. BINOMIAL

• Binomial Setting
• Two discrete outcomes Success/Failure
• Probability of success, p, is same for each
  observation.
• Observations are independent.
• Fixed number of observations, n.

• Geometric Setting
• Two discrete outcomes Success/Failure
• Probability of success, p, is same for each
  observation.
• Observations are independent.
• Variable is the number of observations required
  to get first success.
• Note x0 does not make sense in this setting!

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Calculating Geometric Probabilities

• If X has a geometric distribution with
  probability p of success and (1-p) of failure on
  each observation, the possible values of X are
  1,2,3, If n is any one of these values, the
  probability that the FIRST success occurs on the
  nth trial is

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Sample Distributions 4,6,8,10,20 Sided Dice X
rolls to get 3.
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Geometric Series!

• If you have had Analysis, then you have heard of
  a Geometric sequence. As you (hopefully) recall,
  when the common ratio, r, is between 1,1, the
  sequence is convergent to 0. SO?
• The SUM of that Infinite Series has a clean
  formula a Initial Value, r common ratio
• The initial value of our distribution is p.
  The common ratio is (1-p). SO, we get
• This should make sense, since this is a
  probability distribution and the SUM should be 1!

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Mean of Geometric Distribution

• If X is a geometric random variable with
  probability of success, p on each trial, then the
  expected value of the number of trials required
  to get the first success is
• SO, if we have a 25 chance of success, we should
  EXPECT to get our first success after 1/(.25)
  4 trials.
• How do we interpret this correctly?

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Standard Deviation

• VARIANCE of X is given by the equation
• SO, the Standard Deviation would be

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Example Problem

• An oil exploration firm is to drill wells at a
  particular site until it finds one that will
  produce oil. Each well has a probability of .1
  of producing oil. It costs the firm 50,000 to
  drill each well.
• What is the expected number of wells to be
  drilled?
• What are the expected value and standard
  deviation of the COST of drilling to get the
  first successful well?
• What is the probability that it will take at
  least 5 tries to get the first successful well?
  At least 15?

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Solution to Example

• Expected number SO, 1/.110.
• Expected Cost, would be 50,00010500,000
• Standard Deviation of CostSO, SD of Cost would
  be9.48750,000474,342
• P(X 5) 1- geometcdf(.1,4) .6561
• P(X 15) 1 geometcdf(.1,14) .2288

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Another Formula! ?

• If we want to find the probability that it will
  take MORE THAN n number of trials to get the
  first success, we can use the formula