STA3045F: Integration Summary

IntegrationSummary


Let Cr 4 u be a measurespace
t
f r s IR is a measurable function E MF

Definition of an integral

I f du
s some measure
measurable
function




2
Special Cases of Integration

To define expectation
let be a random variable X
f
let u be a probability measure IP

1 E X Sr X dip

2 Seab
f dat fab
Riemann integrable
foes doc
over bounded
region
Six Em fans
3
f d

, Integral of Indicators
3

Sa du
f Sr f Ia du
Let Ae f

Sef du Sr Ia du

MCA

DarthVader Law

If f e mft
then Snf du Sin Ets t at


Expectation E x S IP Ex t at


I IPx Ct 07 at


Equality of integrals wrt to a measure



If two functions f 9
are f a e
g n
then Snf du In g du