ch3
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ch ch 3 3 Multionari Multionari ate ate Multionariate Random Var Multionariate Random Var iables iables Random Variabl Random Variabl es es
description
Multionariate. ch3. Random Variables. Multionariate Random Variables. 3.1. Cumulative. distribution function. Cumulative distribution function. Definition 3.1. Let S be the sample space associated with a particular. experiment. X and Y be two r.v. assigning to. - PowerPoint PPT Presentation
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ch3ch3 MultionariateMultionariate
Multionariate Random VariablesMultionariate Random Variables
Random VariablesRandom Variables
3.13.1 CumulativeCumulativedistribution functiondistribution function
Cumulative distribution functionCumulative distribution function
Definition 3.1
Let S be the sample space associated with a particular
experiment. X and Y be two r.v. assigning to
a real number vector, (X, Y) , are called
two-dimensional random variable. Denoted by (X,Y)
e
)(eYS
)(eX
S
a) Joint cdf a) Joint cdf
),(),( yYxXPyxF
Definition 3.2
Let X, Y be two random
cumulative distribution
r. v. (X, Y) is defined as
function (cdf) of bivariate
variables. The joint
xo
y),( yx
yYxX ,
),(),( yYxXPyxF
0),(lim),(
yxFyFx
0),(lim),(
yxFxFy
1),(lim),(,
yxFFyx
0),(lim),(,
yxFFyx
(2)(2)
Properties of bivariate cdf F(x,y)
xo
y),( yx
yYxX ,
(1) F((1) F(x,yx,y) is non-decreasing about ) is non-decreasing about xx and and yy. i.e.. i.e.
),(),(, 2121 yxFyxFthenxxif
),(),(, 2121 yxFyxFthenyyif
),()0,(),(lim00
yxFyxFhyxFh
(3) (3) FF((x,yx,y) is right continuous in each argument, i.e.) is right continuous in each argument, i.e.
),(),0(),(lim00
yxFyxFyhxFh
thenyyandxxIf ,2121
);( 2121 yYyxXxP 0),(),(),(),( 11211222 yxFyxFyxFyxF
(4(4))
00
),( 21 yx ),( 22 yx
),( 11 yx ),( 12 yx
2x1x
1y
2y
x
y
)()( xXPxFX ),( YxXP ),(, xF YX
ObviouslyObviously
)()( yYPyFY ),( yYXP ).,(, yF YX
the marginal cdf can be determined by the joint cdf.the marginal cdf can be determined by the joint cdf.i.e.i.e.
b) marginal cdf
Definition 3.3
If FX,Y (x,y) is the joint cdf of the r.v.s X and Y, then
the cdfs FX (x) and FY (y) of X and Y are called
marginal cdfs of X and Y, respectively.
