/m11 answer: You are right that when you generate simulated data, the std. devs will not be exactly 4.0 and 3.0. That's ok. Every simulation only captures some aspects of the real-world phenomenon being studied. It's a judgment call whether the important aspects are being captured. The same is true for every mathematical model: it never captures all aspects of the real world.

Use real-valued ages for mathematical simplicity. Forcing ages to be integers would be an unnecessary layer of complexity.

I've received a lot of feedback saying it would be helpful to extend the deadline, so we'll extend it by one week, to Tuesday January 18. Please use the extra time to review basic statistics and probability!

/m12 Thanks for your answer. I have another question on problem 2. Did you mean the age difference can be negative? Is it x-y or |x-y|? By definition, it seems to be latter one, but it makes the problem much complicated.

Assume that the age difference can be negative, i.e. do not take the absolute value. As you say, this would make the problem much harder. Also, it would be scientifically less interesting, if this was a real-world scenario.

Someone asked me whether instead of Matlab you can use Octave, an open-source alternative. The answer is yes, but I recommend Matlab.

Matlab is expensive, but a full student version is available at the UCSD bookstore for only $100.

Compared to Octave, Matlab is more comprehensive, and more widely used by other researchers. Matlab can increase research your productivity by more than it costs.

I am generally favorable to the open-source philosophy, but I am not a fan of open-source projects that mostly just aim to copy the functionality of existing software products. Doing that seems like a waste of human talent. Innovation should be rewarded and extended rather than copied.

On problem 4, I can see how we can get to a point where the value of investing in the stock market will be above the value of a cash investment with a certain probability.

Selecting m will change that probability. Namely, we can find an m corresponding to the cash investment being k standard deviations from the mean of the stock investment. The question is how many standard deviations, or equivalently, what probability of being better than the cash investment is desired?

This seems like it is dependent on how much you are willing to "gamble". Is it OK to leave the selection of m as a function of k standard deviations or probability?

Also, for part b, I'm a bit confused as to what makes a good price for the oracle. I initially thought that paying something less than the difference between the cash and expected stock market returns would be good, but that doesn't seem ok, since a large percentage of the time, you may make more than the cash but less than the expected value of stock market, resulting in a loss.

Can you say something about how to go about the selection of the appropriate cost, given the method for selecting m I described above. Thanks.

For problem 4, just pay attention to the expected value of your wealth under alternative policies. That is, do not pay attention to risk. From this point of view, a 10% chance of $1000 is worth the same as a 50% chance of $200: you are neither risk-loving (like a gambler) nor risk-averse (like most real investors).

The value of the information is the expected extra profit that the information allows.

For problem 1c, do you mean that we assume p(y|x) = k for y=1 and (1-k) for y=0? or is it something more complicated so that y can be 0 or 1, but dependent on x also?

I'm sorry, but I guess I'm still missing something after your clarification of Problem 4. If we only look at expected values, don't we always expect the stock market outperform the cash investment, since d>c. If we don't care about risk, why wouldn't we always invest in the stock market?

I guess I'm just having trouble formulating the problem so that we can find a optima for m that will correlate to maximium expected wealth or something to that effect.

In problem 4, are we expected to generate an algebraic expression for m where the gain is maximized? That is, something of the form, gain is maximized at m = F(c,d,s), or is it sufficient to give an function, F(c,d,s,m), which should be maximized (presumably numerically) with respect to m?

/19 answer: I just mean that every y value is a member of the set {0,1}. y can be dependent on x. This means that p(y|x1) =/= p(y|x2) in general.

/m20 answer: Yes, if we are risk-neutral and we don't have the oracle, we will always invest in the stock market. However, the answer from the oracle may cause us to invest in cash for the first m years. We will always invest in the stock market for the last n-m years.

The oracle tells us the outcome of the random variable SUM xi for i=1 to i=m. Knowing the actual outcome is better than knowing its expectation!

/m21 answer: An algebraic expression is ideal, but failing that, an expression to be maximized numerically is almost as useful in practice.

In both cases, one or two figures to show qualitatively how m depends on c, d, s^2, and n would be informative.

In probem #2. Can we assume anything about the independence of the ages of the husband and wife?

/m25 answer: You should investigate what follows *if* you assume independence. This is the central point of the problem.

It seems more reasonable to invest in cash for the last n-m years. Say you start investing in stock at 20 to withdraw at 65. At 60 you transfer all assets to cash. For the last five years c works on the higher rate, but shields against drops in the stock market due to high s.

If the opposite strategy is taken, and you start investing in cash at 20 then transfer to stock at 25, c works on a much smaller amount, and large s could make final w< initial w.