[Solved] B555 Written Assignment 2

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 .

• Use the change of variables formulas given in class to calculate the distribution over y.
• What is the range of values of y for which Pr(y) is not zero.
• Verify that Pr(y) calculated in part (1) is normalized; that is, verify that ^R_y 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( = 3x₁ + 2x₂ + 5,² = 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 V ^T. Now consider the optimization problem:

argmax_{x | _xTx1} ^xTSx

that is, we seek a vector x of norm at most 1 maximizing the quadratic form x^TSx. 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.

By using Equations (C.22) and (C.26) prove Equation (C.28) in the textbook