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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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• 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