Download Review Lecture 9 Continuous Random Variables What is a Random

7/23/2014
What is a Random Variable?
Online Review Course of
Undergraduate Probability and Statistics
• Random Variable: a real-valued function of the
outcomes of the experiment
Review Lecture 9
Continuous Random Variables
– Ω→
(maps sample space onto the real numbers)
• Example
– Ω = {all UT students}, X = height of randomly selected
student
Chris A. Mack
• Discrete versus Continuous random variable
Adjunct Associate Professor
– Option 1: round height measurement to the nearest
inch. Result = discrete RV
– Option 2: measure height with infinite precision.
Result = continuous RV.
Course Website: www.lithoguru.com/scientist/statistics/review.html
© Chris Mack, 2014
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© Chris Mack, 2014
Continuous Random Variable
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Probability Density Function
• Consider a discrete RV:
• Definition of PDF:
lim
• Now let the distance between values of x go to
PDF, probability density function
zero
• Probability is the area under the pdf curve
Replace summation with integral
Replace PMF(x) with PDF(x) dx
a
© Chris Mack, 2014
3
• Normalization:
4
Expectation and Variance
0
Discrete:
"# $
1
0
• The probability at a point is zero:
• Example: Uniform probability
a
b
% &# $
! "# $ ' Continuous:
1
!
© Chris Mack, 2014
b
© Chris Mack, 2014
PDF Properties
• Non-negativity:
Probability per
unit length along x
→
"
5
© Chris Mack, 2014
% &
! "# $
'
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7/23/2014
Uniform PDF
Cumulative Distribution Function
• Uniform pdf between a and b, zero outside this
range
(
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% &
5
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2* ! +
! "# $
• CDF for a continuous RV
4
• CDF is monotonically non-decreasing
5
b
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© Chris Mack, 2014
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© Chris Mack, 2014
Normal Distribution
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/
1
1
2
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© Chris Mack, 2014
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• Define >
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9
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© Chris Mack, 2014
10
• How are discrete and continuous RVs
related?
• How are PDF and CDF related for a
continuous RV?
• Know how to calculate expectation and
variance given a PDF.
• What is the normal distribution and some
of its important properties?
Histograms are used to
compare to PDFs
• Wikipedia is a very good source for
distributions and their properties
© Chris Mack, 2014
; <, 7 ' -./CD
Review #9: What have we learned?
• Some common PDFs:
Exponential distribution
Cauchy distribution
Student’s t-distribution
Chi-squared distribution
Log-normal distribution
Beta distribution
Gamma distribution
Etc.
“standard normal”
(z-score)
– Let ? @
A.
– If
; <, 7 ' then ?
There are Many, Many PDFs
–
–
–
–
–
–
–
–
8
:
• A linear function of a normally distributed RV
produces a normally distributed RV
27
7'
8
Normal Distribution
• Also called the Gaussian distribution
1
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11
© Chris Mack, 2014
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