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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, 77, 79
Continuous Random Variables
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For discrete random variables, can assign probabilities to each
outcome in the sample space
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Continuous random variables take on all possible values in an
interval(s)
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Random variables such as heights, weights, times, and
measurement error can all assume an infinite number of values
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Need different way to describe probability in this setting
• 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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0. 0.1 0.2 0.3 0.4
Re la tiv e
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Fr
0. 0.1 0.2 0.3 0.4
Re la tiv e
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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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P ( a  X  b) 
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P( X  x) 
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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
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?
Example (Continuous Uniform)
• Consider the following curve:
• Draw curve:
• Is this a density?
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:
CDF for the Continuous Uniform
Example CDF
• Suppose that X has pdf:
• cdf:
Using the CDF to Compute Probabilities
• Can use cdf to compute the probabilities of intervals…integration
• Can also use cdf:
P ( a  X  b) 