Problem 11.

Let X₁ , X₂ be i.i.d. normal random variables with mean zero and variance 1

Let U= X₁-X₂, V=X₂-X₁. Show that U and V are independent

Problem 12.

Let X_i,i=1,...,n be i.i.d. mean zero and variance 1 random variables

Show that the sample mean and sample variance of all the X's are independent.