Multivariate Random Variables

We can generalize Definition 3.2.2 as follows.

definition 3.2.4 A T-variate discrete random variable is a variable that takes a countable number of points on the T-dimensional Euclidean space with certain probabilities.

The probability distribution of a trivariate random variable is deter­mined by the equations P(X = xbY = yp Z = zk) = р^к, і = 1, 2, . . . , n, j = 1, 2, . . . , m, k = 1, 2, . . . , q. As in Section 3.2.2, we can define

Marginal probability

m q

P(X = *i) = X S P(X = Xit Y =yjt Z= zk), і = 1, 2, …. n.

j= і k= і

Подпись: P(X = Y = yj image030

Conditional probability

image031

if P(Z = zk) > 0

if P(Y = yp Z = zh)> 0.

Definition 3.2.5 generalizes Definition 3.2.3.

definition 3.2.5 A finite set of discrete random variables X, Y, Z, . . . are mutually independent if

P(X = xi}Y = yp Z = zh. . .)

= P{X = Xi)P(Y = jj)P{Z = zk) . . . for all i, j, k,______________

It is important to note that pairwise independence does not imply mutual independence, as illustrated by Example 3.2.5.

example 3.2.5 Suppose X and Y are independent random variables which each take values 1 or —1 with probability 0.5 and define Z = XY. Then Z is independent of either X or Y, but X, Y, and Z are not mutually independent because

P(X = 1, Y = 1, Z = 1) = P(Z = 1 I X = 1, Y = 1 )P(X = 1,Y=1) = P(X = 1, Y = 1) = y4,

whereas

P(X = 1 )P(Y = 1 )P(Z = 1) = %.

An example of mutually independent random variables follows.

example 3.2.6 Let the sample space S be the set of eight integers 1 through 8 with the equal probability of У8 assigned to each of the eight integers. Find three random variables (real-valued functions) defined over S which are mutually independent.

There are many possible answers, but we can, for example, define

X = 1 for і < 4,

= 0 otherwise.

Y = 1 for 3 — і — 6,

= 0 otherwise.

Z = 1 for і even,

= 0 otherwise.

Then X, Y, and Z are mutually independent because P(X= 1,Y= 1,Z= 1) =P(i = 4) = %

= P{X = 1)P(T = 1)P(Z = 1),

P(X = 1, Y = 1, Z = 0) = P(i = 3) = %

= P(X = 1)P(F = 1 )P(Z = 0), and so on for all eight possible outcomes.

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