5.7 Due on 4/25/2008

Difficult

I am not so sure about part (b) of the definition of martingale, why the expectation of future fortune is equal to Sn?  Is it because Sn+1 depends on Sn?  On example (3), how does E(Sn+1|S0, ..., Sn) become E(Sn+Xn+1|Sn)?  Also, on the proof of Theorem 10, I don't understand why the indicators are used here?

Reflective

I am really having difficulties understanding this section.  In the first paragraph of this section, it says that martingales are used extensively in the modern financial mathematics; but I don't see any connections between them.  I feel that maybe because I can't fully understand this section.  I hope I can understand them better after the lecture.

5.6 Due on 4/15/2008

Difficult
In the proof of Theorem (4), I am not sure why "if Sn-S0=k, then r-s=k and r+s=n". Is k an arbitrary number? Also, in example (22), I am not sure how we get to the third step from the second step.

Reflective
This is the first time I learn about random walk, so it's a challenge for me. Although I can't understand the theorems in this section thoroughly, I think it's fun to learn about random walk; also, I think random walk is useful in the practical world.

5.5 Due on 4/13/2008

Difficult

In the proof of Theorem 6, I am not sure why we should assume "the sums are absolutely convergent".  We make that assumption, is it because we want to use the definition 5 in this section?  Also, in the proof, it refers to Theorem 4.3.4, but I couldn't find that theorem.  Is it a typo?

Reflective

Reading this section helps me realize how the conditional expectation is used in solving the probability problems.  Also, the conditional expectation provides us some simplicity in solving the probability problems, which is showed in the solution of example 10.

4.6 Due on 4/10/2008

Difficult
In the solution of example (9), I am not sure why the Chebyshov's Inequality is used to solved this problem. Is it possible to use the other inequalities listed in this section to solve this problem? The Chebyshov's Inequality is used here, is it because the problem states that "X be a random variable such that var(X)=0"?

Reflective
In the solution of example (16), the differential calculus is used to prove that -logx is convex. That's the part that I think it's interesting, because differential calculus was the first or second class we've taken, and I think I will never use that again, but the differential calculus knowledge is actually requared here. So, reading this section remains me once again, that the materials in the field of mathematics are attached to each other.

4.5 Due on 4/8/2008

Difficult

In the definition, I understand how the convergence is defined; and I tried to prove that, but I didn't succeed, so I am curious about how to prove the convergence of the sequence of distribution function.  Also, in example (3), I am not sure how  become as n goes to infinity.

Reflective

Example (2) and example (3) show me how the distribution functions can be used to solve the probability problems in a more simple way.

3.6 Due on 4/6/2008

Difficult

In section 3.6, I tried to do the proofs for Theorem 6 to Theorem 9.  I used the hint from the top of page 91, so I had no problem doing the proofs except for Theorem 9 (Exponential Function).  I tried to do the proof by following the hint on the top of page 91, but I couldn't get the same result as the textbook.  So, can you show me how to prove Theorem 9 in class?

Reflective

As I read through this section, I tried to understand the definitions and the theorems as much as I can; I also tried to memorize all of them, although I am not sure if it's necessary to memorize all of them.  Also, as I read through the solution of the example, I understand that we can use the Multinomial Theorem to do this problem, but I am still a little confused about the solution.

3.4 Due on 4/3/2008

Difficult

In the solution of the example, I can understand almost all the explanations.  However, I am not sure how the series come up for the probability that it's a derangement.

Reflective

Reading the example in this section helped me understand the definition of derangement; it also show me that permutation can be an important tool in probability calculation.