I am trying the following problem:
Let $(X_1, Y_1)\ and\ (X_2, Y_2)$ be random points on the plane such that $X_1, X_2, Y_1, and\ Y_2$ are independent $N(µ, σ^2)$. Let $D^2\ $ denote the squared distance between the two points.
Determine whether or not $D^2\ and\ T =\frac{1}{4} (X1 + X2 + Y1 + Y2)\ $ are independent.
Here I found $D^2\ is\ 2\sigma^2\chi_2^2\ andand\ T\ is\ N(\mu, \frac {\sigma^2}{4}) $ distribution. But I could not understand which path i have to use?