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Statistics 270 - Lecture 12

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Statistics 270 - Lecture 12 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 ... – PowerPoint PPT presentation

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Title: Statistics 270 - Lecture 12


1
Statistics 270 - Lecture 12
2
  • 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

3
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

4
  • 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

5
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6
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7
  • 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

8
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

9
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?

10
Example (Continuous Uniform)
  • Consider the following curve
  • Draw curve
  • Is this a density?

11
Example (Continuous Uniform)
  • In general, the pdf of a continuous uniform rv
    is
  • Is this a pdf?

12

13
CDF
  • Recall the cdf for a discrete rv
  • The cdf for the continuous rv is

14
CDF for the Continuous Uniform
15
Example CDF
  • Suppose that X has pdf
  • cdf

16
Using the CDF to Compute Probabilities
  • Can use cdf to compute the probabilities of
    intervalsintegration
  • Can also use cdf
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