Actuarial Outpost Transformation of Jointly Dist. R.V
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#1
12-11-2018, 01:43 AM
 imrancairo Join Date: Sep 2011 Posts: 18
Transformation of Jointly Dist. R.V

Consider the transformation: Y1=X1−X2, Y2=X1+X2. We wish to find the joint distribution of Y1 and Y2.

We have

x1=y1+y2/2, x2=y2−y1/2 OR v1(y1,y2)=y1+y2/2,v2(y1,y2)=y2−y1/2

i cant understand how y1+y2/2 and y2-y1/2 derived??

Pls. help me best regard to all
Attached Images

#2
12-11-2018, 08:29 AM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,096

It is just regular algebra, solving 2 linear equations in 2 unknowns. For the first, just add the two equations (add left sides and right sides).

Y1=X1−X2
Y2=X1+X2

Y1+Y2 = 2 X1

X1 = (Y1+Y2)/2

(The way you wrote it was confusing without parentheses, but the picture clearly shows Y1 and Y2 each divided by 2.)

Similarly for X2; subtract the equations.
#3
12-11-2018, 11:56 PM
 imrancairo Join Date: Sep 2011 Posts: 18

Thanks Sir!