- Solve problem 3.3 (page 174) in the textbook.
- 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.

- Consider a bi-variate normal variable
*X*distributed N(0*,I*) and a univariate*Y*where*Y*|*X*is distributed as N(*µ*= 3*x*_{1 }+ 2*x*_{2 }+ 5*,σ*^{2 }= 25). Calculate an explicit form for*p*(*X*|*Y*= 4) using our template for Bayes theorem for Gaussians. Are*x*_{1}*,x*_{2 }still independent after*Y*is observed? - Consider a real-valued symmetric matrix
*S*with eigen decomposition*S*=*V*Λ*V*. Now consider the optimization problem:^{T}

argmax_{{}*x *| _{x}*T**x*≤1} ^{x}^{T}Sx

that is, we seek a vector *x *of norm at most 1 maximizing the quadratic form *x ^{T}Sx*. What is the optimal solution

*x*?

**Hint: **can you express *x *in the basis formed by *V *?

- Solve problem 3.7 (page 175) in the textbook.
- Solve problem 3.11 (page 175) in the textbook.

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

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