Question

Suppose that X and Y are independent standard normal random variables. Show that U = }(X+Y) and V = 5(X-Y) are also independent standard normal random variables.

Answer 1

the solution. This is infact true for any linear combination of two independent Normal Random variable .

that, Here given. X~ N(0,1), and Mr N/0,1). V=x+y =) -> X2 X = ut but V= x-Y 키 발 (X,Y) (0,r). 2 Ni- 151 X,Y EIR - -| U, VER. 131 1- 2. Now Toint pof of X and y 2 x,yt R. fry (x,y)= I e- x + y 271 -((utva So ne), 2 e 2 f) I 2T 2 42 tuz e 271,032 -14 2 5) leven z 12 e UVER. Waa2 So, clearly, Ioint pdf of u and u can be written in product of two pofs and hence, v and v are independent! hods and also both pofs suggests, "=X+Y NNCO, 1) and x = x 4 WN(0, 1) V2

That is , say if X and Y are two independent standard Normal Variable, then aX+bY and aX-bY are also independent random Variables following normal, where a and b are any real constants, here in this problem, if we take a=b=1/√2 , then we can say that (X+Y)/√2 and (X-Y)/√2 follows standard normal independently. Elaborate solution using transformation is given in the image attached. Thank you.