Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
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<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