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Lecture

Riemann Integration I.pdf

6 pages76 viewsFall 2012

Department
Statistics
Course Code
STAT312
Professor
Douglas Wiens

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123
25. Riemann Integration I
Riemann integration. First consider : [ ]
R, a bounded function. Consider the approximat-
ing histogram ( ), with breaks at {=0
1··· =}and heights {(1) ( )},
where is any point in [ 1]. A rst approx-
imation to the area under is the Riemann sum
( ) = X
=1 ( )
where = 1. Now let the norm =
max ( ) shrink to 0. If ( ) has a limit as we
do this, we call this limit the Riemann integral of
, and say that is Riemann-integrable on [ ]
( []).
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124
−1.96 0.00 1.96

 
−1.96 0.00 1.96

 
     
      
Approximations of area under the Normal density
( ) between -1.96 and 1.96 using Riemann sums
(= areas under histograms); target value = .95.
Each of the points (*) was randomly chosen
in [ 1]; these intervals each have width
= (2 ·1 96 ) and then
( ) = P( ) = 3 92 ·1P=1 ( ).
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