Post on 22-Dec-2015
Chapter 4
Joint Distribution &
Function of rV
Joint Discrete Distribution
Definition
Xxxxx
xXxXxXPxxxf
XXX
k
k
kkk
k
of ),...,,( valuespossible all
,...,,,...,,
be todefined is ),...,,(X variablerandom discrete
ldimensiona- theof pdf)(joint function density y probabilitjoint The
21
221121
21
1),...,,(.2
,...,, valuespossible allfor 0,...,,.1
: satisfied are properties following theifonly if ),...,,(X rV
valued- vectorsomefor pdfjoint theis ,...,,function A
1 2
21
2121
21
21
x x
k
kk
k
k
xxxf
xxxxxxf
XXX
xxxf
Joint discrete CDF
kkk
k
xXxXxXPxxxf
XXX
k
,...,,,...,,
be todefined is ),...,,(X variablerandom discrete
ldimensiona- theof cdfjoint The
221121
21
Example for joint distributions
Consider the following table:
Using the table, we have
Y=0 Y=3 Y=4
X=5 1/7 1/7 1/7 3/7
X=8 3/7 0 1/7 4/7
4/7 1/7 2/7
pX
pY
.7/37/27/1)4(p)3(p 3YP
7/3)5(p7XP
2/7p(5,4)p(5,3)3Y7,XP
YY
X
The Marginal PDF
1
2
2122
2111
21
2121
,)(
,)(
are and of spdf' marginal then the
),...,,( pdfjoint thehas rV discrete of pair theIf
x
x
k
xxfxf
xxfxf
XX
xxxf,XX
Example : Air Conditioner Maintenance– A company that services air conditioner
units in residences and office blocks is interested in how to schedule its technicians in the most efficient manner
– The random variable X, taking the values 1,2,3 and 4, is the service time in hours
– The random variable Y, taking the values 1,2 and 3, is the number of air conditioner units
Expected Values for Jointly Distributed Random VariablesExpected Values for Jointly Distributed Random Variables
Let X and Y be discrete random variables with joint probability density function p(x, y). Let the sets of values of X and Y be A and B, resp. We define E(X) and E(Y) as
For the random variables X and Y from the previous slide example,
).(pE(Y) and )(pE(X)B
YA
X yyxxyx
.7
47
7
48
7
35E(X)
.7
11
7
24
7
13E(Y)
• Joint p.d.f
• Joint cdf
Y=number of units
X=service time
1 2 3 4
1 0.12 0.08 0.07 0.05
2 0.08 0.15 0.21 0.13
3 0.01 0.01 0.02 0.07
1
07.0...08.012.0
i j
ijp
43.0
08.015.008.012.02,2
F
Find E[X] and E[Y] !!
Previously Example– Marginal p.d.f of X
– Marginal p.d.f of Y
3
11
( 1) 0.12 0.08 0.01 0.21jj
P X p
4
11
( 1) 0.12 0.08 0.07 0.05 0.32ii
P Y p
Joint continuous distribution
),...,,( allfor
,...,,,...,,
: as written becan
CDFjoint that theasXsuch of pdfjoint thecalled
,...,,function a is thereif continuous be tosaid is
),...,,(X rV valued- vectorldimensiona-A
21
212121
21
21
k
k
x x
kk
k
k
xxxx
dtdtdttttfxxxF
xxxf
XXXk
k k
kA k
k
dxdxdxxxxfAXP
XXXk
2121
21
,...,,
: have A weevent ldimensiona-k a and
),...,,(X rV valued- vectorldimensiona-A
Th
1),...,,(.2
,...,, valuespossible allfor 0,...,,.1
: satisfied are properties following theifonly if ),...,,(X rV
valued- vectorsomefor pdfjoint theis ,...,,function A
21
2121
21
21
k
kk
k
k
xxxf
xxxxxxf
XXX
xxxf
Ex
10,10,4),(by given is pdfjoint that theAssume 212121 xxxxxxf
a. Then find the joint CDF !
5.0
2 Find b. 21 XXP
The Marginal Continuous PDF
12122
22111
21
2121
,)(
,)(
are and of spdf' marginal then the
),...,,( pdfjoint thehas rV scontinuuou of pair theIf
dxxxfxf
dxxxfxf
XX
xxxf,XX k
Find )( and )( 2211 xfxf From previously example
Ex
)(xf
xxxcxxxf
XXX
33
321321
321
findthen
10,,,
form theof pdfjoint a with continuous be ,,Let
Independent rV
k
iiikkk
ii
k
bXaPbXabXaf
ba
XXX
11111
21
,...,
every if
t independen be tosaid are ,,, Variables Randomt Independen Def.
)()(,,
)()(,,
: holds properties following ifonly if
t independen be tosaid are ,,, Variables Random Th.
111
111
21
kkk
kkk
k
xfxfxxf
xFxFxxF
XXX
Conditional pdf
Joint MGF
0h somefor and ,..., where
)(
: be todefined is existsit if of MGFjoint The
1
1
1
hthttt
eEtM
,...,XXX
ik
Xt
X
k
k
iii
Group
Discuss the exercise bellow !
BAIN, page : 166-
No 7, 9, 21, 30
Time : 30’