hello group,

bold are vectors, unbold scalars.

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

isp(y)=p((x-b)/a)/|a|

I would like to know the above theorem whenaandbare constant vectors but X is a scalar random variable

GivenY=aX+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.

LetY=(Y1,Y2,Y3),a=(a1,a2,a3), andb=(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?