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Transformation of variables in joint distributions

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Transformation of variables in joint distributions
PROBABILITY
Finally we note that, by analogy with the single-variable case, the characteristic
function and the cumulant generating function of a multivariate distribution are
defined respectively as
C(t1 , t2 , . . . , tn ) = M(it1 , it2 , . . . , itn )
and
K(t1 , t2 , . . . , tn ) = ln M(t1 , t2 , . . . , tn ).
Suppose that the random variables Xi , i = 1, 2, . . . , n, are described by the PDF
f(x) = f(x1 , x2 , . . . , xn ) = N exp(− 21 xT Ax),
where the column vector x = (x1 x2 · · · xn )T , A is an n × n symmetric matrix and N
is a normalisation constant such that
∞ ∞
∞
f(x) dn x ≡
···
f(x1 , x2 , . . . , xn ) dx1 dx2 · · · dxn = 1.
−∞
∞
−∞
−∞
Find the MGF of f(x).
From (30.142), the MGF is given by
M(t1 , t2 , . . . , tn ) = N
t2
where the column vector t = (t1
we begin by noting that
···
∞
exp(− 21 xT Ax + tT x) dn x,
(30.144)
tn )T . In order to evaluate this multiple integral,
xT Ax − 2tT x = (x − A−1 t)T A(x − A−1 t) − tT A−1 t,
which is the matrix equivalent of ‘completing the square’. Using this expression in (30.144)
and making the substitution y = x − A−1 t, we obtain
M(t1 , t2 , . . . , tn ) = c exp( 12 tT A−1 t),
where the constant c is given by
(30.145)
c=N
∞
exp(− 21 yT Ay) dn y.
From the normalisation condition for N, we see that c = 1, as indeed it must be in order
that M(0, 0, . . . , 0) = 1. 30.14 Transformation of variables in joint distributions
Suppose the random variables Xi , i = 1, 2, . . . , n, are described by the multivariate
PDF f(x1 , x2 . . . , xn ). If we wish to consider random variables Yj , j = 1, 2, . . . , m,
related to the Xi by Yj = Yj (X1 , X2 , . . . , Xm ) then we may calculate g(y1 , y2 , . . . , ym ),
the PDF for the Yj , in a similar way to that in the univariate case by demanding
that
|f(x1 , x2 . . . , xn ) dx1 dx2 · · · dxn | = |g(y1 , y2 , . . . , ym ) dy1 dy2 · · · dym |.
From the discussion of changing the variables in multiple integrals given in
chapter 6 it follows that, in the special case where n = m,
g(y1 , y2 , . . . , ym ) = f(x1 , x2 . . . , xn )|J|,
1206
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