Random Vectors Sometimes we may have to be concerned with Random experiments whose out - comes will have 3 or more simultaneous numerical charecteristics. (iii) P[(X1,X2. .... Xn) ED] = J J ... J f(xX2. ....Xn) dx, dx2 ... dxn To study the outcomes of such random experiments we require knowledge of n - dimensional random variables or random vectors. where D is a subset of the range space Rn. For example, The marginal pdf of any subset of the n RVs X1,X2. .... Xn is obtained by "integrating out" the variables not in the subset. The location of a space vehicle in a cartesian co - ordinate system is three - dimensional random vector. For example, if n = 3, then Most of the concepts introduced for the two - dimensional case can be extended to the n - dimensional one. fx, ( x ] ) = -00 - 00 Definitions : A vector X : [X1,X2, ..., Xn] whose components X, are RVs is called a random vector. (X1, X2. ..., Xn) can assume all values in some region Ry, of the is the marginal pdf of the one - dimensional RV X, and n - dimensional space. R, is called the range space. The joint distribution function of (X1,X2, ..., Xn) is defined as fx, x, (x1 . X2 ) - | f(x1,x2, X3) dx3 - 0o F(X1, X2. ..., Xn) = P(X1 5 X],X2 S X2 . . .., Xn SXn) is the marginal joint pdf of the two - dimensional RV (X], X2). The joint pdf of (X1,X2, ..., Xn) is defined as The concept of independent RVs is also extended in a natural way. an f(x1X2. ...Xn) - ax, . dx2 .". OXn F(x1, X2, ..., Xin) The RVs (X1,X2, ..., Xn) are said to be independent, if and satisfies following conditions. f(x1,X2. ...,Xin) = fx, ( x1) . fxz (x2) ."fxn ( Xn) (i) f(x1, X2, ....Xn) 2 0, for all (X1.X2. ..., Xin) The conditional density function are defined as in the following examples. If n = 3, then (it) . . f( xx2.. In ) dx, dxz " dxn = 1 Rn f(x1, X2/X3) f(X1, X2,X3) fx, (x3) and fix1/(x2,x3)} = f(x1, X2, X3) fx2 Xz (X2, X3)