Chapter 17 Random Variables576
ThenH 1 is independent ofM, since
PrŒMD1çD1=4DPr
MD 1 jH 1 D 1
DPr
MD 1 jH 1 D 0
PrŒMD0çD3=4DPr
MD 0 jH 1 D 1
DPr
MD 0 jH 1 D 0
This example is an instance of:
Lemma 17.2.1.Two events are independent iff their indicator variables are inde-
pendent.
The simple proof is left to Problem 17.1.
As with events, the notion of independence generalizes to more than two random
variables.
Definition 17.2.2.Random variablesR 1 ;R 2 ;:::;Rnaremutually independentiff
for allx 1 ;x 2 ;:::;xn, thenevents
ŒR 1 Dx 1 ç;ŒR 2 Dx 2 ç;:::;ŒRnDxnç
are mutually independent.
17.3 Distribution Functions
A random variable maps outcomes to values. Often, random variables that show up
for different spaces of outcomes wind up behaving in much the same way because
they have the same probability of taking any given value. Hence, random variables
on different probability spaces may wind up having the sameprobability density
function.
Definition 17.3.1.LetRbe a random variable with codomainV. Theprobability
density function (pdf)ofRis a function PDFRWV!Œ0;1çdefined by:
PDFR.x/WWD
(
PrŒRDxç ifx 2 range.R/;
0 ifx...range.R/:
A consequence of this definition is that
X
x 2 range.R/
PDFR.x/D1:
