X, Y are independent. g(x) and h(y) are independent for any function g, h, not depending on X, Y.

x: sample mean of X₁, which has distribution N(0, 1)

s²: sample variance

xand s² are independent

x= U₁ = (X₁ + X₂)/2, U₂ = X₁ - X₂

For n = 2, show that U₁ and U₂ are independent