Let $ X, Y $ be independent normalized normalized random variables and $ X, Y neq 0 $. Find the density of $ frac {X ^ {2}} {Y ^ {2} + X ^ {2}} $

I was given the tip of the first density calculation of $ (X ^ {2}, Y ^ {2}) $ then calculate the density of $ ( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2}) $

When I follow the advice: I know that $ X ^ {2} $~$ Gamma ( frac {1} {2}, frac {1} {2}) $ and $ Y $ also. In addition, $ X ^ {2} $ and $ Y ^ {2} $ are still independent. Therefore, the density $ f _ {(X ^ {2}, Y ^ {2})} (x, y) $ can write $ f_ {X ^ {2}} (x) f_ {Y ^ {2}} (y) $ or $ f_ {X ^ {2}} $ and $ f_ {Y ^ {2}} $ are the density functions of $ X ^ {2} $ and $ Y ^ {2} $

My next idea, with the above advice in mind is to consider a card $ varphi: (x, y) mapsto ( frac {x} {x + y}, x + y) $

It ensues while $ ( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2}) = varphi (X ^ {2}, Y ^ { 2}) $

and $ f _ { frac {X ^ 2} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2}} (a, b) = f _ { varphi (X ^ {2}, Y ^ {2})} (a, b)

$

Note that $ varphi ^ {- 1}: (a, b) mapsto (ba, b-ba) $ And so $ det D varphi ^ {- 1} (a, b) = det begin {pmatrix} b & a \

-b and 1-a

end {pmatrix} = b (1-a) + ab implies det D varphi ^ {- 1} ( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2}) = (Y ^ {2} + X ^ {2}) (1- frac {X ^ {2}} {Y ^ {2} + X ^ {2} }) + ( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}) (Y ^ {2} + X ^ {2}) = Y ^ {2} + X ^ { 2} $

And so $ P _ {( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2})} (A) = int_ {A} f_ {( frac {X ^ {2}} {Y ^ {2} + X ^ {2}}, Y ^ {2} + X ^ {2})} (x, y) dxdy = int _ { varphi ^ {- 1} (A)} f _ {(X, Y)} (x, y) times (X ^ {2} + Y ^ {²}) dxdy = int _ { varphi ^ { – 1} (A)} f_ {X} (x) times f_ {Y} (y) times (X ^ {2} + Y ^ {²}) dxdy $

Where should I go from here?