probability density function of a vector

hello group,

bold are vectors, unbold scalars.

I know that is *a* and *b* are nonzero scalars and X is a random variable with pdf *p(x)*, then the pdf of Y=*a*X+*b*

is *p(y)=p((x-b)/a)/|a|*

I would like to know the above theorem when **a** and **b** are constant vectors but X is a scalar random variable

Given **Y**=**a**X+**b**, find p(**Y**)?

I assuumed that this formula that holds for scalars will work for vectors too

p(**Y**)=Int[-inf,inf]d(**y**-(**a***x-**b**))p(x)dx

where d is the dirac delta, then I dont know if this is the appropiate way to interpret the dirac-delta in this problem.

Let **Y**=(Y1,Y2,Y3), **a**=(a1,a2,a3), and **b**=(b1,b2,b3)

d(**y**-(**a***x+**b**))=d(y1-(a1*x+b1))d(y2-(a2*x+b2))d(y3-(a3*x+b3))

then using the property of the dirac-delta:

d(f(x))=Sum{i}d(x-xi)/|f'(xi)|

where xi are the zeros of f(x), f(xi)=0

Let f(x)=y1-(a1*x+b1), f'(x)=a1

xi=(y1-b1)/a1, only one zero

I put it back to get this result:

p(**y**)=d(y2-(a2*xi+b2))d(y3-(a3*xi+b3))p(xi)/|a1|, is it correct?