Statistics 270 - Lecture 12

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Statistics 270 - Lecture 12
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• Last day/Today More discrete probability
  distributions
• Assignment 4 Chapter 3 5, 7,17, 25, 27, 31, 33,
  37, 39, 41, 45, 47, 51, 65, 67, 77, 79

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Continuous Random Variables

• For discrete random variables, can assign
  probabilities to each outcome in the sample space
• Continuous random variables take on all possible
  values in an interval(s)
• Random variables such as heights, weights, times,
  and measurement error can all assume an infinite
  number of values
• Need different way to describe probability in
  this setting

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• Can describe overall shape of distribution with a
  mathematical model called a density function,
  f(x)
• Describes main features of a distribution with a
  single expression
• Total area under curve is
• Area under a density curve for a given range
  gives

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• Use the probability density function (pdf), f(x),
  as a mathematcal model for describing the
  probability associated with intervals
• Area under the pdf assigns probability to
  intervals

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Example

• A college professor never finishes his lecture
  before the assigned time to end the period
• He always finishes his lecture within one minute
  assigned end of class
• Let X the time that elapses between the
  assigned end of class and the end of the actual
  lecture
• Suppose the pdf for X is

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Example

• What is the value of k so that this is a pdf?
• What is the probability that the period ends
  within ½ minute of the scheduled end of lecture?

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Example (Continuous Uniform)

• Consider the following curve
• Draw curve
• Is this a density?

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Example (Continuous Uniform)

• In general, the pdf of a continuous uniform rv
  is
• Is this a pdf?

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CDF

• Recall the cdf for a discrete rv
• The cdf for the continuous rv is

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CDF for the Continuous Uniform
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Example CDF

• Suppose that X has pdf
• cdf

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Using the CDF to Compute Probabilities

• Can use cdf to compute the probabilities of
  intervalsintegration
• Can also use cdf