statistics question ??

[Sinistral]

I have another doubt: In the following ques (image attached) is there any direct way to solve it .



I did repeated substitutions (transformations),
1. A= X-Y, B=X+Y to get f(a,b)
2. C=A/B , D=B to get f(c,d)
3. U=C^2 , F=D to get f(u,f)
4. marginal pdf of f(u)
eventually F(1,1) is the distribution which i got and it is the correct ans. but is there any shorter or quicker  way to solve this? because in the hints it's given -- [chi square(1)] / [chi square (1)] = F(1,1)

[Amit Goyal]

Since X and Y are independent standard normal. Both (X-Y)/srqt{2} and (X+Y)/sqrt{2} are standard normals. Square of standard normal is Chi square (1). And ratio of two independent Chi-square(1)s is F(1,1). Therefore, you just need to show that X-Y and X+Y are independent. And then you can use the hint.

srqt{2} denotes square root of 2.