Let x (n × 1) be a standard normal random vector.

(a) What is the joint density function of the m-vector y = Ax + b where A (m × n)

is a fixed matrix with rank m and b is a fixed vector?

(b) Consider the vectors u = Ax and ???? = Bx where A (m × n) and B (p × n) are fixed matrices. Show that if AB′ = ????, then u and ???? are distributed independently of each other.

(c) Consider in particular the cases A = n−1????′ where ????′ denotes the row n-vector of ones, and B = In − n−1????????′ (n × n). Show that these matrices satisfy the condition in part

(b) and interpret the result.

(d) Does it follow from the result in part

(c) that the sample mean and sample variance of a normal random sample are independent random variables? Explain.