4.4, 4.9(c) due on 10/31

Difficult
I was able to follow the coin example up until equation 11. Why is E(X) = E(X')? I'm guessing because their probabilities are equal, so does that mean that knowing the first toss is head makes no difference? Also, is the problem asking for the expected value of the length of runs of HEADS or TAILS? I assume we are looking at runs of heads out of convenience, but I'm not sure. The computation of the variance also confused me. I think it's because I haven't practiced it as much, but it was hard for me to follow.
Reflective
Aside from the above confusion, I found section 4.4 fairly straightforward since most of it relates to what we've already learned about conditional probability. It's nice to get a review of the material and also see it being applied in a different way. In section 4.9, I thought it was a little tricky since you had to think about writing p as np/n and realize that (1+ x/n)^n equals e^x. Otherwise, that example was also straightforward.

4.10 due on 10/29

Difficult
Once again, I'm having trouble applying the formula. In part (b), I kind of followed along, but what happens to the k! in the denominator? Is it because the book only considered the cases where k=0 or k=1, since then k! would be 1? Also, to find the number of bites a random postman sustains, do we take the derivative? If so, why?
Reflective
I was very happy that I could understand the derivation of the Poisson distribution in part (a). The book is hard to follow, but after playing with the equation a bit I got the final formula. However, as mentioned above, I'm still confused about which values are assigned to which variables when applying a formula.

4.3 due on 10/19

Difficult
I'm not sure how we can assign positive infinity or negative infinity to E(X) if by definition, E(X) must be finite. I'm also not sure what the purpose is of the real-valued function g(.). Is the domain of g the actual random variable or is it the range of X? What does it mean to to find the expected value of X^2? What is the definition of a moment? I'm also very confused by the Coupons example, as well as others in this section. I think the reason I'm having so much trouble following these examples is because I still don't completely understand the previous section.
Reflective
The idea of expectation makes a lot of sense to me. Since more weight is given to a particular value of X based on its probability, the greater the probability, the more likely it is that value occurs. Pictorially it's like a graph, where the value with the highest expectation is where the area below is the largest.

4.2 due on 10/17

Difficult
This reading introduced a few new concepts, leaving me with lots of questions. First off, I understand the definition of a probability mass function but I'm having trouble applying it. I'm running into the same problem as before: I don't know where the numbers come from. In the poker example, I know the numerator is all the outcomes which satisfy 2 pairs, but I don't know how to get that numerator. I also thought the equations for a proper random variable and the Key Rule looked similar. What's the difference between the two? And what's an example of where the sum of probability mass functions is less than one?
I also did not fully understand the Poisson Distribution and the Negative Binomial Distribution. For Poisson, I don't see how the sum equals 1. I referred back to Theorem 3.6.9, but that only confused me more. For the Negative Binomial, how do we decide what f(r) equals? And how do we know f(r) lies within [0,1]? The cumulative distribution function F(x) completely confused me. I don't understand what it does or its relationship to f(x). Figure 4.1 didn't really help either. Is F(x) like finding the integral of f(x)?
Reflective
Overall, I feel like the book is skipping steps and I'm having trouble filling in the missing pieces. I feel like I should be able to figure out some of these steps on my own but I can't. I had the most trouble in this reading with the cumulative distribution function. I want to say that F(x) is like finding an integral since it's a distribution, but I'm still not sure.

4.1 due on 10/15

Difficult
This reading was not as difficult as the last couple of readings. The part I had the most trouble with was the definition of a discrete random variable. At first, I thought X(w) was a probability function instead of a function mapping Omega to a countable set of real numbers. What I don't understand is why the sample space doesn't need to be countable. From the Darts example, when w is an event where the dart doesn't hit the board, then X(w) = 0. Does that mean that even though the sample space is uncountable, the only set of outcomes that matter is when the dart hits the board, which is countable?
Reflective
The concept of an indicator reminded me of what I've learned in my programming classes. It's like an indicator is a "true or false" test, with 1 being true and 0 being false.

3.3, 3.6(9)-(12), 3.12 due on 10/12

Difficult
Theorem 3.3.3 confused me a little because I'm not sure how we derived the equation. It seems we take the product of (ni+1) instead of only (ni) because we're adding any number of symbols from zero to ni, which equals (ni +1). However, I don't understand why we also subtract 1. I also don't understand how to derive the exponential function theorem and the binomial theorems.
Reflective
I did not find Example 3.12 helpful at all. I feel like I'm missing the reasoning behind the problem. As I've said before, it's nice to see examples of how and when to apply certain theorems, but not knowing why does not make me feel confident when approaching homework problems. As mentioned in my first blog post, grasping new math concepts is still something that I need to work on.

3.1, 3.2 due on 10/10

Difficult
I understood most of section 3.1 because a lot of it I've learned before. However, the following definition confused me: "a number n (say) of objects or things are to be divided or distributed into r classes or groups." In the example following this definition, there wasn't any reference to this definition or what n and r correlate to in a given problem. In section 3.2, this was cleared up since theorems (1) and (2) gave better explanations of how we can split n objects into r groups. Although, theorem (3) gave me some trouble because I don't quite follow the proof and I'm not sure what a multinomial coefficient is.
Reflective
The example in section 3.2 really helped me understand how the multinomial coefficient works, even though I'm still not sure what it is. It seems to me if you have Mn (x, y, z) then that means there are n total objects, with x number of objects of type 1, y number of objects of type 2, and z number of objects of type 3. The reason you divide n! by the product of (x! y! z!) is in order to have no repeats in the permutations.

2.2, 2.13 due on 10/8

Difficult
I had trouble understanding the definition of conditionally independent and pairwise independent. It seems conditionally independent means two events are still independent of each other regardless of another third event. For pairwise independence, I thought of it as an extension of an independent collection, where the finite set F has cardinality 2. I also had trouble with the idea of protocol. It seemed like the first example (Tom and siblings) was like the ones we've done in class. It didn't occur to me that Tom could be a twin or a girl. For the goat/car example, I'm still not sure how the emcee's involvement changes the probability.
Reflective
Despite my confusion, I find the idea of protocol very intriguing. It shows we have to consider all factors of a problem or question before thinking about the answer. It was interesting to note how the examples in section 2.13 were published and the majority of the readers (including myself) were deceived and answered incorrectly.

2.1, 2.7 due on 10/5

Difficult
I understand definition 2.1.1 and theorem 2.1.2, but the poker revisited example completely throws me off because I don't know how they got the numbers. What I mean is, how do you come up with 1/(52C5) as the probability for the intersection of R and SA or (51C4)/(52C5) as the probability of SA? I thought the probability of drawing the ace of spades is 1/52. I also had trouble with repellent and attractive events. The second example using Bayes's Theorem made no sense to me. Like in the poker problem, I don't know how they got certain values for the probabilities.
Reflective
This reading has showed me that I need to review combinations. I believe that is the main reason why I am so confused on the poker example. This reading has also pointed out that I'm able to understand concepts, but when I try to apply the formulas, I'm not sure which equation to use or when to use it.

1.8, 1.12 due on 10/3

Difficult
I didn't find this reading really difficult to understand, but there were some parts that were confusing when I first read it. The example in section 1.12 threw me off for a little bit. I tried to think about how I would go about solving the problem on my own before reading the solution below. Part (a) was pretty easy, but I was stumped by parts (b) and (c). I had to think about why we could consider some births "fictitious" when dealing with families with less than three children. If the solution was not given to me, I'm not sure I would have figured that out by myself.
Reflective
I like examples because I find them very useful. This reading cleared up how to apply some of the concepts we've learned in the previous readings. As mentioned above, I found section 1.12 very helpful. Section 1.8 was also good because I got to see a case in which using the complementary event is more efficient (faster) than a more direct approach.