Question

In: Math

Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2...

Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y = (Y1,Y2,Y3)′ using : Multivariate normal distribution properties.

Expert Solution

given that

X1,X2 and X3 be iid N(0,1)

hence

Y1 = X1 + X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3

adding Y1,Y2 and Y3

Y1+Y2+Y3=X1+X2+X3+X1-X2+X1-X3=3X1

this gives

now

Y2 =X1-X2 gives

Y3=X1-X3 gives

now jacobian is given by

=1/3 [(-2/3)(-2/3)-(1/3)(1/3) -1/3[(1/3)(-2/3)-(1/3)(1/3)]+1/3 [(1/3)(1/3) -(1/3)(-2/3)]

=(1/9) +(1/9) +(1/9) =3/9 =1/3

now joint pdf of X1 ,X2 and X3 is given by

now joint pdf of Y1,Y2 and Y3 is given by

f(y1,y2,y3) =f(x1(y1,y2,y3),x2(y1,y2,y3),x3(y1,y2,y3)) *|J|

milcah answered 1 year ago

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