AP Statistics

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AP Statistics
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Binomial and Geometric Distributions

• Chapter 8

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Definitions of a Binomial and Geometric
Distributions

• Definition of a Binomial Distribution
• The distribution of the count X of successes in
  the binomial setting is the Binomial
  Distribution.
• X is B(n,p)
• n is the number of observations.
• P is the probability of each of the observations
• Definition of a Geometric Distribution
• The distribution of the count n of trials in the
  geometric setting is the Geometric Distribution.
• X is G(n,p)
• n is the number of observations.
• P is the probability of how long the success
  would take to happen once.

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Definitions of Settings

• The Binomial Setting
• Each observation falls into one of just two
  categories, which for convenience are called
  successes and failures
• There is a fixed number of n observations
• All observations are independent.
• The probability is the same for each observation.

• The Geometric Setting
• Each observation falls into one of just two
  categories, which for convenience are called
  successes and failures
• All observations are independent.
• The probability is the same for each observation.
• The variable of interest is the number of trials
  required to obtain the first success.

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Binomial Distributions

• A Binomial Distribution has a probability
  distribution functionor a Binomial PDF.
• The Binomial PDF is located on a TI calculator
  under 2nd / DISTR / 0binompdf.
• This is used to find the binomial probability of
  a value X. So, it is used to find an exact match
  to a number X.

• A Binomial Distribution has a cumulative
  distribution function or a Binomial CDF.
• The Binomial CDF is located on a TI calculator
  under 2nd / DISTR / Abinomialcdf.
• This is used to find the binomial probability of
  a value X. So, it is used to find the added
  probabilities up to a number X.

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Binomial and Geometric PDF and CDF

• Binomial PDF is written as binomialpdf(n, p, x)
• n is trials
• p is probability
• x is number you are looking for
• Binomial CDF is written as binomialcdf(n, p, x)
• n is trials
• p is probability
• x is number you looking for up to that number

• Geometric PDF is written as geometricpdf(n, p, x)
• n is number of trials required to reach one
  success
• p is probability
• x is number of successes needed
• Geometric CDF is written as geometriccdf(n, p, x)
• n is number of trials required to reach one
  success
• p is probability
• x is number of successes needed

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Important Notes

• The approximation is based on finding an area
  under the normal curve that approximates the
  y-coordinate of the appropriate binomial
  probability plot.
• In order to use the normal approximation to
  answer binomial questions the binomial is
  discrete and the normal is continuous, the values
  of interest must sometimes be adjusted.
• When n is large, it is unnecessary to apply the
  continuity correction for an interval of values
  since the error introduced by not applying it is
  very small.

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A Few Shortcuts and Hints

• 1-seq (x, x, 0, 10, 1) L1
• 2-binompdf (10, .1, L1)
  L2 will calculate each P(x )

3- Turn off any Y stuff 4-STAT PLOT
histogram X list L1 Y list L2 5- Set window
X-.5, 10.5 Y-.1, .45

• L1 L2 L3
• .34868
• .38742
• .19371
• .0574