Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
<strong>1.</strong>8. RANDOM VARIABLES<br />
(IR,B) in (IR,B). Then,<br />
∫ ∫<br />
(g ◦X)dP =<br />
Ω<br />
IR<br />
gdP X .<br />
Remark <strong>1.</strong>9 Let F X be the d.f. associated with the probability<br />
measure P X . The integral<br />
∫ ∫ +∞<br />
gdP X = g(x)dP X (x)<br />
IR<br />
will also be denoted<br />
∫<br />
as<br />
gdF x =<br />
IR<br />
−∞<br />
∫ +∞<br />
−∞<br />
g(x)dF X (x).<br />
Proposition <strong>1.</strong>19 IfX isanabsolutelycontinuousr.v. withd.f.<br />
F X <strong>and</strong>p.d.f. withrespecttotheLebesguemeasuref X = dF X /dµ<br />
<strong>and</strong> if g is any function for which ∫ IR |g|dP X < ∞, then<br />
∫ ∫<br />
gdP X = g ·f X dµ.<br />
IR<br />
In the following we will see how to calculate these integrals for<br />
the most interesting cases of d.f. F X .<br />
IR<br />
(a) F X discrete: The probability is concentrated in a finite or numerablesetD<br />
X = {a 1 ,...,a n ,...},withprobabilitiesP{a 1 },...,P{a n },.<br />
Then, using properties (a) <strong>and</strong> (b) of the Lebesgue integral,<br />
∫ ∫ ∫<br />
gdP X = gdP X + g(x)dP X<br />
IR D X D<br />
∫<br />
X<br />
c<br />
= gdP X<br />
D X<br />
∞∑<br />
∫<br />
= gdP X<br />
=<br />
n=1<br />
{a n }<br />
∞∑<br />
g(a n )P{a n }.<br />
n=1<br />
ISABEL MOLINA 42
Change language