Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

134 Multivariate Distributions

(x 1 ,x 2 ) (0,0) (0,1) (0,2) (1,1) (1,2) (2,2)

p(x 1 ,x 2 ) 121 122 121 123 124 121

Findp 1 (x 1 ),p 2 (x 2 ),μ 1 ,μ 2 ,σ 12 ,σ 22 ,andρ.

2.5.11.Letσ 12 =σ 22 =σ^2 be the common variance ofX 1 andX 2 and letρbe the
correlation coefficient ofX 1 andX 2. Show fork>0that

P[|(X 1 −μ 1 )+(X 2 −μ 2 )|≥kσ]≤

2(1 +ρ)

k^2

2.6 ExtensiontoSeveralRandomVariables

The notions about two random variables can be extended immediately tonrandom
variables. We make the following definition of the space ofnrandom variables.

Definition 2.6.1.Consider a random experiment with the sample spaceC.Let
the random variableXiassign to each elementc∈Cone and only one real num-
berXi(c)=xi,i=1, 2 ,...,n. We say that(X 1 ,...,Xn)is ann-dimensional
random vector.Thespaceof this random vector is the set of orderedn-tuples
D={(x 1 ,x 2 ,...,xn):x 1 =X 1 (c),...,xn=Xn(c),c∈C}. Furthermore, letAbe
a subset of the spaceD.ThenP[(X 1 ,...,Xn)∈A]=P(C),whereC={c:c∈
Cand(X 1 (c),X 2 (c),...,Xn(c))∈A}.

In this section, we often use vector notation. We denote (X 1 ,...,Xn)′by the

n-dimensional column vectorXand the observed values (x 1 ,...,xn)′of the random

variables byx. The joint cdf is defined to be

FX(x)=P[X 1 ≤x 1 ,...,Xn≤xn]. (2.6.1)

We say that thenrandom variablesX 1 ,X 2 ,...,Xnare of the discrete type or

of the continuous type and have a distribution of that type according to whether

the joint cdf can be expressed as

FX(x)=

∑

···

w 1 ≤x 1 ,...,wn≤xn

∑

p(w 1 ,...,wn),

or as

FX(x)=

∫x 1

−∞

∫x 2

−∞

···

∫xn

−∞

f(w 1 ,...,wn)dwn···dw 1.

For the continuous case,

∂n

∂x 1 ···∂xn

FX(x)=f(x), (2.6.2)

except possibly on points that have probability zero.
In accordance with the convention of extending the definition of a joint pdf,
it is seen that a continuous functionfessentially satisfies the conditions of being
apdfif(a)fis defined and is nonnegative for all real values of its argument(s)