### Problems to work for Week 5

See (2.282) for the definition of a symmetric Levy-stable random variable.

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Use the projection formula (3.64) in analogy to the arguments leading to (3.74) to work out the time-scaling properties of a process with symmetric Levy-stable increments.

See (2.282) for the definition of a symmetric Levy-stable random variable.

I suggest you work these problems to check your understanding. These are appropriate for the material covered in Week 4. I will post problems for Week 5 tomorrow.

I will mark your solutions if you hand them in, but the scores will not count towards your grade.

I will mark your solutions if you hand them in, but the scores will not count towards your grade.

- Verify that the equations (2.176-7) in M for the gaussian copula in fact define a density by numerically integrating it over the domain for various values of the correlation paramter. Note that the integrand diverges in the corners, so you may want to use the
`eps`constant in MATLAB to truncate the integration bounds. You may also want to plot the function, for example using`ezsurfc()`. - Extend the regression analogy by using the equation (2.165) in M for the conditional mean of a normal dependent variable, X3, given values X2=x2 and X1=x1 for
__two__normal (but possibly correlated) independent variables. Interpreting this as a point estimator, compare what you get to the formulae for generized least squares.

I have posted two versions of the solutions I worked, one in Mathematica 6.0 and the other in MATLAB R2007b. If you need me to re-work the Mathematica solution for a previous version, please let me know. In both cases, I used trial and error to identify that the optimal portfolio was 55% risky asset and 45% riskless asset.

The first assignment is available. Please work this problem on your own and write up the solution. I will collect it next Wednesday, September 26.

Please contact me before Sunday if you wish to meet.

**No office hours next Tuesday.**

As was announced last week, until you hear otherwise assume that the Wednesday evening sessions will start at the classroom in FordH 127 at 5:00 and move to lab in LindH 024 after the break at 6:30.

The lab is scheduled to be open to the public prior to the lecture, so it should still be open after the break. If not, I can open it.

Here is the diagram that I shared in class from Casella & Berger that shows the relationships between the major classical distributions.

The files we use in the lab such as the M-file for `binomial()` will be kept in the directory http://www.math.umn.edu/~dodso013/fm503/docs/.

It is not required for success; but I encourage you to learn and use "functional" programming techniques in MATLAB (or Mathematica or whatever high-level programming language you use), especially when working with random variates.

In contrast to "procedural" programming, functional programming generally allows you to avoid loops or superfluous explicit index variables, and is therefore more compact to code and quicker to execute.

You will need to be learn to think in terms of generic functions, called variously "pure" functions, "lambda" functions or "anonymous" functions. Examples of commands in MATLAB that avoid loops and indexes are `fplot()` and `arrayfun()`. In order to perform a loop, use the range operator, such as `1:100` in MATLAB, to quickly generate an array of indexes.

You can load a sample of random variables into MATLAB using the command

sscanf(urlread('http://www.math.umn.edu/~dodso013/fm503/case1.dat'),'%f')

These are drawn uniformly between zero and some unknown upper bound. Please provide an interval estimate for the upper bound.

The objective is to have the narrowest interval that includes the true value. This is a contest; the winning team members will take home a Sven & Ole's bumper sticker!