# STA414H1 Quiz: hw1

## Document Summary

Let x, y be two independent random vectors in rm. (a) show that their covariance is zero. (b) for a constant matrix a rm m, show the following two properties: E(x + ay ) = e(x) + ae(y ) Var(x + ay ) = var(x) + avar(y )at (c) using part (b), show that if x n ( , ), then ax n (a , a at ). Here, you may use the fact that linear transformation of a gaussian random vector is again gaussian. Answer the following questions: (a) can a probability density function (pdf) ever take values greater than 1? (b) let x be a univariate normally distributed random variable with mean 0 and variance. Let x, y rm and a rm m. Suppose that x rn m with n m and y rn, and that. We know that the maximum likelihood estimate of is given by. In this problem, you will write a function that performs.