1. Solve problem 3.3 (page 174) in the textbook.
2. Consider a multivariate random variable (of dimension 2) uniform[1,2]2 and
the random variable y define as .
(1) Use the change of variables formulas given in class to calculate the distribution over y.
(2) What is the range of values of y for which Pr(y) is not zero.
(3) Verify that Pr(y) calculated in part (1) is normalized; that is, verify that Ry Pr(y)dy = 1.
3. Consider a bi-variate normal variable X distributed N(0,I) and a univariate Y where Y |X is distributed as N(µ = 3x1 + 2x2 + 5,σ2 = 25). Calculate an explicit form for p(X|Y = 4) using our template for Bayes theorem for Gaussians. Are x1,x2 still independent after Y is observed?
4. Consider a real-valued symmetric matrix S with eigen decomposition S = V ΛV T. Now consider the optimization problem:
argmax{x | xTx≤1} xTSx
that is, we seek a vector x of norm at most 1 maximizing the quadratic form xTSx. What is the optimal solution x?
Hint: can you express x in the basis formed by V ?
5. Solve problem 3.7 (page 175) in the textbook.
6. Solve problem 3.11 (page 175) in the textbook.
7. By using Equations (C.22) and (C.26) prove Equation (C.28) in the textbook.
