Jhan Math 3c Blog

12.6.2 Part 2

2006-12-03T18:57:00.000-08:00

Section 12.6.2 Part 2 Blog Entry

Having read section 12.6.2 part 2 in the text:

Difficult:
On page 895 example 8, I could not see how the quantity (Sn-np)/(square root of (nsigma^2)) [being approximately standard normally distributed), could result in the new quantity square root (n) multipled by (Xn-p)/(square root of (p(1-p))) when both the numberator and denominator are divided by n. I see how the text substituted the value for the var(x) and then substituted the value for sigma once again in the original equation, but I didn't get how dividing the numerator by n could reslt in Xn-p multiplied by square root (n) on top.

Reflective:
I thought it was rather interesting how this section relates and connects to the idea of normal distributions that was covered in the last section 12.5. These types of problems and examples in this section seem to use the same general methodology for solving normal distribution problems in terms of referring to the z-score tables but adds the extra application of using the form given in the central limit theorem.

12.6.2 Part 1

2006-12-03T18:55:00.000-08:00

Section 12.6.2 Part 1 Blog Entry

Having read section 12.6.2 part 1 in the text:

Difficult:
It says on page 892 that the central limit theorem does not give a good approximation. Why does it not give a good approximation?

In example 5, for the correction to get the better approximation for the probability, do we always use a difference of 0.5 for each side? Does it vary or is it always given?

Reflective:
I don't understand exactly what the purpose of the central limit theorem is quite yet. It seems at this point as if we are just adding on to what we have learned during the normal distribution section. It seems a bit straightforward in terms of plugging into the formula, but I think it may be rather difficult to choose which way is most efficient to solve a certain problem, since it seems like the problems given are similar to ones that are possible to solve through other methods as well.

12.6.1 Part 2

2006-11-26T15:17:00.000-08:00

Section 12.6.1 Part 2 Blog Entry

Having read section 12.6.1 part 2 in the text:

Difficult:
In example 2 on page 890, I was not sure about the statement included in the text that reads, "we want to investigate how fast Xn(bar) converges to p". When I read the problem/question, I don't think I caught that this was what the problem was asking for. Which statement in the original question implies that we should be looking for how quickly Xn(bar) converges to p?

I wasn't sure how we could make the substitution for var(Xi) with p(1-p) in the solution to example 2 on page 890.

Reflective:
The examples in this last part of section 12.6.1 were very difficult for me to follow personally. When looking through the steps of each solution/example, it took me a while to understand what steps the text had taken to arrive at the next line of the example. I realized that a lot of this section involved notation manipulation, substitution, and other ideas that were supposed to simplify the initial problems to help you get to the answer.

12.6.1 Part 1

2006-11-26T12:45:00.000-08:00

Section 12.6.1 Part 1 Blog Entry:

Having read section 12.6.1 part 1 in the text:

Difficult:
On page 888, I was actually a bit unsure in the beginning parts of the proof of Chebyshev's Inequality. I was a bit confused on how we know that the E(X - mean)^2 =sigma^2 is less than infinity. I was a bit caught off guard with the text's mention of expectations in this section since I was actually trying to figure out and make sense first of the probability notations that it had given us just above it. Because I am not quite sure how this expectation of (X-mean)^2 came about, I was not able to fully grasp the idea/equation directly below in the proof on page 888 that states that P((X-mean)^2 greater than or equal to c^2) is less than or equal to E(X-mean)^2/c^2 = (sigma^2)/(c^2).

Reflective:
I wasn't quite sure th purpose of the text's explanation of the change in notation of Markov's Inequality. Given the Inequality, we could see that by multiplying both sides by the denominator (of the right side of the inequality), a, you could get the conclusion on page 888 that states that the EX is greater than or equal to "a" multiplied by the probability of random variable X greater than or equal to "a". It seemed somewhat unnecessary to include the mathematical explanation of this change in notation.

12.5.4 Part 2

2006-11-26T12:17:00.000-08:00

Section 12.5.4 Part 2 Blog Entry:

Having read section 12.5.4 part 2 in the text:

Difficult:
I don't think I had any specific questions for the text examples, but I encountered some confusion while solving or looking at some of the assigned homework problems and I wasn't quite able to figure them out by referring to the book examples. Just in general, is there a possibility that answers like 0.0015% are possible in these types of survival rate and hazard rate problems?

Also, in the actual text on page 882, I was wondering how the two laws figured into the sample problems. Although it says in the examples to "use Gompertz Law and Weibull Law" with hazard functions, etc., but the actual laws don't seem to provide any steps or help in solving the problems since the hazard functions are already given in a usable form in each of the corresponding problems.

Reflective:
I think out of all the sections we have covered thus far, this 12.5.4 section has the most correlation with real life situations since it deals with radioactive decay, among other life form characteristics. I think it is rather interesting that such imperfect lifeform characteristics and trends can be equated using set formulas. These formulas most likely do not account for any variations or errors, but it is still interesting that these formulas could be derived and applied in the first place.

12.5.4

2006-11-21T03:04:00.000-08:00

Section 12.5.4 Blog Entry

Having read section 12.5.4 in the text:

Difficult:
On page 876 example 15, I think I was able to understand and follow with the mathematical calculations of determining the probabilities of the organism living more than 100 years and more than 150 years, but I couldn't quiet grasp what the text meant by this organism not aging. How and why is the nonaging property a characeristic feature of the exponential distribution?

In example 18 on page 879, I was not quite sure how the inverse of the F(x) function played into the equation.

Reflective:
The first couple of examples in this section were very straightforward and easy to follow, howeer, example 18 was rather tricky for me because of all of the substitution and multiple variables that appeared throughout the example problem. I thought it was rather interesting to delve more closely into the idea of decay and growth using the exponential function. Prior to this section and course, it seemed as if all the textbooks only gave the equation for radioactive decay, but this section helped to explain and break down this idea so that we could see how it could be broken up to be better understood.

12.5.3

2006-11-19T17:36:00.000-08:00

Section 12.5.3 Blog Entry

Having read section 12.5.3 in the text:

Difficult:
I actually did not understand example 12 on page 874. Prior to this example, I followed the calculations in deriving the EU (mean) and variance, but in this example, in trying to figure out the probability of U less than or equal to 0.3, I tried to plugging the values into the equation given on page 872 P(U within (x1, x2))= (x2-x1)/(b-a). But if the values of x1 and x2 are 0 and 1, then what are the values of b-a and how does this equal 0.3? As a result, I was not able to understand how the text wrote the successive values of the uniform random variable (0.2859, 0.9233, 0.5187, 0.8124, and 0.0913).

Reflective:
Personally, the concepts in this section were particularly difficult for me to understand. There weren't many examples (for each action/problem type) and just few examples in general in this section, which I think made understanding the ideas even more difficult, since there weren't more practice problems/examples to refer to. I am not completely sure what a uniform distribution is at this point, and also am a bit confused in how to interpret it on a graph like the one on the bottom of page 874. I was able to fully comprehend deriving the mean and variance, but other than that, it feels as if I am missing something in trying to understand the other concepts in this uniform distribution section.

12.5.2 Part 2

2006-11-12T19:43:00.000-08:00

Section 12.5.2 Part 2 Blog Entry:

Having read section 12.5.2 part 2 in the text:

Difficult:
On page 870 example 10, I wasn't quite sure why P(Z greater than z) is equal to 1-F(z). I thought that the notation 1-F(z) was used to denote F(-z), yet there is no F(-z) in the problem.

Although I read the section explaining the central limit theorem, I still can't quite understand what it is or how you apply it in problems.

Reflective:
I think this part of the section was a bit more straightforward and more or less involved substituting values for the mean and standard deviation into the already derived equations for F(z), F(x), etc. I think it would be very helpful to draw simple sketches of the normal distribution/density curves and label some points for the mean, since it would help to visualize the areas that would need to be calculated. I think it would provide a rather reliable "common sense" check of sorts when solving problems in this section.

12.5.2 Part 1

2006-11-12T19:41:00.000-08:00

Section 12.5.2 Part 1 Blog Entry:

Having read section 12.5.2 part 1 in the text:

Difficult:
I did not understand or follow how the text was able to derive/get the equation of F(x) = P(X less than or equal to x)= the integral from negative infinity to x of the integrand in terms of z on page 864 above example 5. Originally, the equation for a normal distribution was in terms of x and the e was raised to the power of -(x-mean)^2/2(sigma)^2 but now, the sigma is gone from the coefficient as well as the e power.

Reflective:
Personally, it is rather interesting to encounter these concepts and the idea of normal distributions during this calculus class, because I covered this material briefly during statistics class. However, although the general topics are similar, the application and methodology in approaching normal distributions are very different. Adding a calculus twist to the normal curves and normal distributions is interesting since it gives us another way of solving or looking at the same problem. I am hoping that encountering this information in statistics class will prove to be an adequate foundation for the more difficult and complex applications and ideas that I will see in the future sections.

12.5.1

2006-11-07T21:48:00.000-08:00

Section 12.5.1 Blog Entry

Having read section 12.5.1 in the text:

Difficult:
Conceptually, I don't really understand how in example 3, the expectation/mean is equal to the integral of xf(x)dx from negative infinity to infinity can also be equal to the integral of the density function of a random variable X given by f(x)={3x^2 for x in which x is greater than 0 but less than 1 and 0 otherwise. The integral's intervals of integration go from 0 to 1 and I can't really see how it can be said that it is the equivalent of the integral evaluated from negative infinity to infintity.

In example 4, what exactly is r? Is it just a variable that is commonly denoted by x since the variable x takes on a different role/value later on in the proof/example?

Reflective:
I was a bit taken back at the sight of all the integrals and application of integral calculus in this section. However, it was a bit refreshing to use the concept of integrals again when calculating the area under the curve of the probability distribution function.

12.4.6 part 2

2006-11-05T01:53:00.000-08:00

Section 12.4.6 (Part 2) Blog Entry:

Having read section 12.4.6 in the text:

Difficult:
In the long proof for the Poisson Approximation to the Binomial Distribution, I can't quite grasp how the text is changing/manipulating the limit equation after bringing down the n exponent (due to the ln properties) in order to make the limit approach 0/0 or infinity/infinity. I understand the logic behind rewriting the coefficient n into the denominator as 1/n as well as multiplying by 1 in the form of x^n/x^n, but how did the text deduce the equation in that line to simply the limit of ln(1+y)/y as y-->0? It seems like they didn't account for the x^n that was multiplied, especially if y is a substitution for x^n/n in the numerator and deniminator already.

In example 36, is there a specific reason why the k values given/chosen for calculation are only 0, 1, and 2?

Reflective:
The proof was a bit extensive and at times a bit difficult to see explicitly what you had to do next. Regardless, I thought it was pretty interesting that in comparison to the other sections on the probability of picking certains colored balls from an urn, choosing a certain card from a deck, and flipping a certain outcome on a coin, this section seems to deal more with application to real world/life science models or situations. It seems pretty interesting that researchers have deduced certain formulas and equations to study the trends and spreads of certain diseases, populations, etc.

12.4.6

2006-11-02T12:49:00.000-08:00

Section 12.4.6 Blog Entry

Having read section 12.4.6 in the text:

Difficult:
I don't really understand the second line of the last part of Example 34 in which the text was equating the mean or EX of the random variable of a Poisson distribution. I think I was caught a bit off guard by the text's method of splitting up the lambda^k in the numerator into a lambda coefficient and a lambda^(k-1) in the numerator. But then in the next step, I don't really understand how the fraction was deduced again into lambda^k devided by k! which is then equal to e^lambda. I'm wondering if it's just a common sense/exponent rule that I have forgotten.

When computing the variance, how do you know to start off with E[X(X-1)]? Is this the equivalent of the other form var(X)= E(X^2) -(EX)^2?

When computing the variance, why does the lower starting value of the sum k(k-1)P(X=k) change from k=0 to k=2 and then back to k=0? It doesn't seem like a typo since the same line (except with the difference in k=0 and k=2) is written twice and said to be equal to one another.

Reflective:
I found the last parts of Example 34 rather interesting. The text explained how to derive the mean and variance of the Poisson distributed X random variable, and in the second line, the test split up and factored out the lambda^k by dividing it up into lambda coefficient and a lambda^(k-1) which remained in the numerator. It is a clever yet still tricky way of going out manipulting the equations to derive the mean/EX. Personally, I don't think I would have even thought of going about solving the problem in this manner.

12.4.5

2006-10-28T22:56:00.000-07:00

Section 12.4.5 Blog Entry

Having read section 12.4.5 in the text:

Difficult:
I understood most of the steps involved and the solving of Example # 31 on page 840, but I did not really grasp why the problem was given. What kind of difference would the clause "following n unsuccessful trials" have on the original "probability of no success in k trials"? And more importantly, I don't quite understand the idea that the event {X is greater than n+k} is in the event {X is greater than n}. How do we draw this conclusion to make sense of and compute the conditional probability? For clarification, in one of the first lines of the examples, when it states, "To compute the conditional probabiliity ...we use... and then it states formulas, does the numerator of the fraction P(X greater than n+k, X greater than n) basically denote the equivalent of the numerator of the definition of the conditional probability from the previous section in which P(A l B)=P(A intersection B)/ P(B)? So basically, does the comma that is used in the numerator sort of signify the intersection symbol from before?

Intuitively, I could figure out that the chance of getting an asymptomatic child was approximately 3/4, but in the actual example #32, I am not quite sure why P(X greater than 1)=P(X greater than 4 l X greater than 3).

Reflective:
I was a rather amused and surprised when I saw the concept of conditional probabilities appear spontaneously in Example 31 on page 840 of Section 12.4.5. It was interesting to be able to apply the previously learned topic into another completely different idea. I think sifting through and understanding each of the lines of the derivations for EX/mean and variance were a bit overwhelming and tedious. Would we have to know how to derive these formulas in the blue box on page 842(which basically sums up and just lists the formulas that the previously listed steps amount to)?

12.4.3

2006-10-22T20:18:00.000-07:00

Section 12.4.3 Blog Entry

Having read section 12.4.3 in the text:

Difficult:
I was a little confused on page 832 when the text explained that "as all outcomes ar equally likely, we find P(S(5-subscript) =3) ...." if S(n-subscript)=number of successes in n trials. Since the definition of S(n-subscript) was given first, I understood it to be true, but when looking at the example, it said that out of five trials, there were three successes (or 1's). If this is true, then shouldn't S have a 3-subscript rather than a 5-subscript?

On page 833, in example 20, I was a bit confused as to why ther (n-k) exponent was not accounted for in the solution. I think since (1-p) is raised to the (n-k) power in the formula, the (1- .5) should be raised to the (.5-3) power in the solution.

Reflective:
It kind of felt as if this section on binomial distribution should have been included before the previous section about mean and variance and discrete distributions. It seems more pertinent to the probabilities, combinations, and permutations section in the previous section 3 of chapter 12. I think this section 12.4.3 was much more straightforward than the previous two sections since it involved applying and using the given definitions and equations of the section to each individual problem.

12.4.2

2006-10-16T23:08:00.001-07:00

Section 12.4.2 Blog Entry:

Having read section 12.4.2 in the text:

Difficult:
It states on page 824 that it is important not to round to the nearest integer in some cases since it can lead you to draw incorrect and inaccurate conclusions. But, I was just wondering, how many decimal places should answers usually be rounded to?

After reading the paragraph description about the blue box rule for any random variable on page 826 which states "let a and b be constant. Then E(aX+b) = a(EX)+b and var(aX+b) =(a^2)var(X) I still cannot figure out exactly what this statement meants and how these two sides of the equation are equivalent.

On page 830, when the text discusses the idea of independence using the new notations for the random variables, do the notations for when X and Y are discrete random variables P(X=x,Y=y) = P(X=x)P(Y=y) imply an intersection between X and Y variables as it is written in the definition of independence?

Reflective:
In this section, I was a bit surprised at the terminology and vocabulary. I have encountered and applied the ideas of standard deviation, variance, averages, relative frequencies, etc, but I have never actually heard or used words like expected value (mean). I think knowing the different names or labels for certain values or concepts will be rather useful and a good thing to keep track of, since in other situations, either of the terms could appear. I have dealt with variance before, but the notations in this particular formula/definition were a bit new and foreign to me (page 825 Definition of variance).

12.4.1

2006-10-16T23:08:00.000-07:00

Section 12.4.1 Blog Entry:

Having read section 12.4.1 in the text:

Difficult:
I don't understand why in the second example on page 819 in section 12.4.1 Discrete Distributions, it states that X: Omega (Sample Space) --> R takes on values 1, 2, or 3. Did the text just set R as these random values or is there a reason that these three numbers were chosen? Why is it that Y(H)=1, Y(TH)=2, and Y(TTH)=3? I understand the first example for X(HHH)=3 or X(TTH)=1 or X(TTT)=0, since the number signified the number of heads (H) appeared in the coin tosses, but I do not understand the relationship in the second example.

For the definition of a probability mass function, I was wondering if there was an upper bound for the second condition on page 820. The second condition of the definition is the summation of p(x) from x is equal to 1, where the sum is over all values of X with P(X=x) is greater than 0. Is there an upper bound for this summation of p(x)?

I was wondering where the text derived the statement or assumption on the bottom of page 820 in the "solution" section. The text states that "The function F(x) is defined for all values of x included in R. For instance, F(-2.3) =P(X less than or equal to -2.3)=P(empty set)=0 or F(1)=P(X less than or equal to 1)= P(X= -1 or 0) =0.3. I cannot seem to find a reason for the text to make the statement/assumption that F(-2.3)=P(empty set)=0. Where did they get this from?

Reflective:
I found this section to be rather difficult to comprehend. I did not expect piecewise functions and limit concepts to appear in this section/chapter after all of the probability concepts were covered earlier. I think I still have a difficult time in understanding exactly what the distribution function and probability mass function are for. I think after reading I was able to understand the inner workings and individual steps of how they arrived at some of the numbers using the corresponding values on the charts, but I can't seem to grasp exactly why they would use these methodologies.

12.3.4

2006-10-12T11:56:00.000-07:00

Section 12.3.4 Blog Entry:

Having read section 12.3.4 in the text:

Difficult:
I was wondering if on page 813, if there was a reason why the subscript was changed from an "i" to a "j." Since the start of section 12.3.4 The Bayes Formula, the text had written the definition of conditional probabilities P(Bi l A) = P(A intersection Bi) / P(A) and P(A intersection Bi) = P(A l Bi) P(Bi) in terms of the subscript Bi with i-1,2,.....n. But in the next line, equation (12.17) in the law of total probability, P(A) = sum of P(A l Bj) P(Bj) from j=1 to n is written in terms of j. Is there a specific reason for this? Does it make a difference? In the Bayes Formula, both variables i and j are used in the equation. What does j represent?

On page 814, I was a bit confused by the statement, that "the probability is quite small that a person is infected given a positive result," in the HIV test example. Shouldn't the probability be big that a person is infected given a positive result? I think looking at the numbers make sense in the above lines but the logic doesn't seem to make sense to me.

Reflective:
I think this section was a bit more straightforward than the other sections. Although I found some of the text and descriptions to be a bit wordy (with possibly some typos or mistakes, although I am not completely sure), I think the concept of Bayes formula was understandable and conceivable. This section was rather short, and I think the previous section also discussed the similar formula and ideas. I think this section dealt more with applying the concepts in the previous section to examples in real life such as the HIV/AIDS idea.

12.3.3

2006-10-10T16:42:00.000-07:00

Section 12.3.3 Blog Entry

Having read section 12.3.3 in the text:

Difficult:
On page 811, I wasn't sure why the text put such emphasis on making sure that both (12.13) and (12.14) must hold in order to show that three events A, B, and C are independent. According to the previous two examples 5 and 6 and the blue definition box above example 5 on page 810, that "two events A and B are independent if: P(A intersection B) = P(A) P(B). I thought that saying "P(A intersection B) = P(A) P(B) P(A intersection C) = P(A) P(C) P(B intersection C) = P(B) P(C)" in (12.13) was the exact same as saying "P(A intersection B intersection C) = P(A) P(B) P(C)" in (12.14). Is there a difference? Is that why the text wants us to make sure we understand that BOTH conditions must be present for the events to be proven independent of one another?

On the top of page 812, I wasn't sure what the text was saying about the number of conditions that followed each of the various numbers of sets: "When there are three sets, we just saw that there are four conditions--namely, there are (3 choose 2) conditions involving pairs of sets and (3 choose 3) conditions involving all three sets. With four sets, there is a total of (4 choose 2) + (4 choose 3) + (4 choose 4) =11; with five sets, 26 conditions; and with ten sets, 1013 conditions." I can see that when there is a pair (events A and B), you only have to check "P(A intersection B) = P(A) P(B)" as the formula states and I can see that the logic also works with three sets, but once the text begins talking about 4 and 5 sets, I can't seem to figure out how they came up with all the 11 combinations, etc. When I tried to think about and write out the conditions for the four sets example, I think I could only come up with about 7 rather than 11.

Reflective:
I think my biggest problem during this section (12.3 in general) is getting used to the notations and determining whether I am showing the correct steps/work for each of the problems. Personally, I feel that I am having a bit of trouble writing down/choosing variables for the probability problems and choosing the proper equations to use. I think thinking first about which equations/formulas/methods to use is a bit confusing for me, and I realized that just reading the assigned math problem and using elementary steps to first figure out what the problem is saying/asking for (tree diagrams, etc.) and then using the probabilities and fractions to get the answer seemed a bit easier to understand in my mind than using the actual formulas. I am hoping that if I practice enough, I will get used to writing down the formulas and applying them to the correct types of problems in the future.

12.3.2

2006-10-07T21:29:00.000-07:00

Section 12.3.2 Blog Entry

Having read section 12.3.2 in the text:

Difficult:
In the very beginning of section 12.3.2 The Law of Total Probability, I was a bit confused on what the notation sample space omega = U from i=1 to n of Bi meant. It was labeled as (ii) on page 808. Do this just mean that each value of Bi from i=1 to i=n is the partition and you would add all of these up to form the sample space?

Reflective:
Once again, I think the use of multiple variables and notations made the material a bit confusing to understand and figure out in the beginning. When I first read the bottom of page 808 about letting A be an event and writing A as a union of disjoint sets using the partition of the sample space, I did not really understand how (A intersection B1) U (A intersection B2) U....U (A intersection Bn) could equal the same A, but when I looked at Figure 12.15, I think I was able to visualize the equation/problem better and understand what it meant. I think it was helpful for me to observe and think about the diagrams and figures that accompanied each example in this section. Also, in example 3, at first I was really confused at what the text was trying to determine because I don' t think I understood that it was making a distinction between testing positive for a disease because you actually have the HIV virus or because your test was misread also taking into account the actual prevalence of the disease in a population.

12.3.1

2006-10-05T23:19:00.000-07:00

Section 12.3.1 Blog Entry

Having read section 12.3.1 in the text:

Difficult:
On page 808 example #2, I did not quite understand why the text gave an explanation and made a distinction about the difference between using 12.8 or 12.9. Although I realize and agree with the authors that it is easier to begin with the first draw, it seems like the text is just stating the obvious since the only reason that the idea of a conditional probability exists is since the first event impacts the next event.
Q: Are conditional probabilities only possible when an experiment or event happens without replacement or when events are not independent with respect to one another?
Q: When given the equation P(A l B) = P(A intersection B) / P(B), it seems as if the example (and any other cases that come to mind) will only use the rearranged form of P(A intersection B) = P(B l A) P(A). Does the original form of the equation get used as well in some examples/cases or do we usually end up solving for the second rearranged form of the equation more often?

Reflective:
This text deals with the idea of conditional probability in a way that was a bit foreign to me personally. I have solved similar probability problems in the past that deal with not replacing the item that was removed in the first draw/turn, but I have no recollection of the symbols or notations like P (A l B) or the term conditional probability. I think this new notation made me think that the topic was more confusing than it actually was once I read through and solved through the examples in this section. I think compared to the previous probability problems in which replacement was used, these types of conditions and cases seem more relevant to real life models since data is most always variable and subject to change which could cause a conditional probability to be used.

12.2.2

2006-10-04T00:16:00.000-07:00

Section 12.2.2 Blog Entry:

Having read section 12.2.2 in the text:

Difficult:
On page 801 at the very top of the page, I was not exactly sure why the statement "B=B1 union B2 with B1 intersection B2 equal to an empty set" is true. I think I can understand that the first part of the statement is true since B1 and B2 are the only options or choices for picking green balls in two picks, but I am not quite sure why the intersection of the two is an empty set. If B1 = {exactly one ball is green} and B2 = {both balls are green}, do they not have anything in common?

I also found the Mark-Recapture Method on page 802 example 12 about fish to be a bit confusing and wordy in its explanation. I am not sure how the equation l A l = (K k) (N-K n-k) allows us to find the total number of ways of obtaining a sample size n with exactly k marked fish. Specifically, I do not understand the meaning of the subtractions N-K and n-k, but I can see the use of combinations and the purpose of the coefficient (K k) to select the k marked fish from the total K fish.

Reflective:
Once again, I think that learning and remembering the proper notations and symbols such as the difference between omega and the absolute value of omega were important in this section. I found it a bit confusing at first to understand the difference between A and l A l when trying to read through and solve the examples at the beginning of this section, but I think I was slowly able to adjust as I continued through the section's reading. I originally thought that this class would be more oriented and directed towards calculus. It was somewhat refreshing and interesting to see that some ideas from calculus such as the idea of not being able to differentiate across discontinuous functions such as the ratio pN / P N-1 on page 803. I also had the most trouble with understanding the maximum likelihood estimate during this particular section of the text, but I think as I attempted to read it multiple times and wrote the steps of the process out, it became a bit more clear where the values and equations were coming from, although I am still a bit shaky on the purpose and meaning of example 13 and the maximum likelihood estimate.

12.2.1

2006-10-02T00:44:00.000-07:00

Section 12.2.1 Blog Entry

Having read section 12.2.1 in the text:

Difficult:
On page 796, when discussing the topic of Basic Set Operations, I do not understand why the text makes a note that the sample space of Ai from i=1 to n is equal to A1 U A2 U ...U An = (A1 U A2 U... U A n-1) U An. I am not sure why the unions of the sample space subsets from A1 to A n-1 are separated on the right side of the equation signs in the last part of the statement. [this was also done for the example about the intersections of subsets of Ai.] Is this distinction made only to show that the pairing or ordering is not significant in these cases?

Referring to page 797 and the definition of a probability, I was a bit confused looking at condition #2 where the P(empty set) =0 and P (finite sample space) = 1. I do not understand why the probability of an event A which in this case is the finite sample space is equal to 1 rather than less than or equal to 1 (as in the above condition). Does this condition basically state that it is impossible to have no value/empty set and the probability must be a finite number/exist? Also, for condition #3 of the definition of a probability, wy is it that "for two disjoint events A and B, P(A U B) = P(A) + P(B)? If you must use addition to calculate the probabilities of event A and event B to calculate the probability of the union of event A and event B, isn't it possible that the probability could result in a number greater than 1 (which would conflict with the idea that the probability must stay within the numbers 0 and 1?

Shouldn't the second equation in the solution be P{4,5} = P(4) + P(5) rather than 3 and 4? [example 6, page 798]

Reflective:
I think the most confusing aspect of this section was remembering the terminology and classifications of the basic set operations. During high school math courses, these ideas were covered rather briefly and I do not remember solving many problems that required actual application of this material, so I think it is crucial to be able to identify the different methods and operations for this section. Although some of the examples only dealt with flipping coins, I think that the topic of probability will come up in real life when trying to conduct research experiments, studying general populations of people, learning about genetics and diseases, and more.

12.1.1-12.1.4

2006-09-30T13:53:00.000-07:00

Section 12.1.1- 12.1.4 Blog Entry

Having read sections 12.1.1-12.1.4 in the text:

Difficult:

One thing that I was not sure about was why 0! is equal to 1. How can we prove that 0! = 1 or is this statement a set idea/value that if assumed allows the rest of the factorials to be calculated? Also, how are the binomial coefficients (n k) = (n n-k) [couldn't figure out how to use superscripts in this blog] as stated on page 789? From the way that the text refers to the identity, it seems as if the binomial coefficients (n k) = (n n-k) can be equal only when k=0, but I'm not sure when it would equal each other in different cases. On page 790, the text refers to applying the counting principles to expand the binomial theorem (x+y)^n but I wasn't quite sure (and couldn't quite remember) how to write the expanded form when given values of n and k. When looking at the solutions section of the example, I was a bit confused when the text stated "A term of the form x^k . y^(n-k) occurs (n k) times since there are (n k) ways of selecting the factor x k times from among the n facotrs (x + y)."

Reflective:

In all of my previous math classes, I never felt as if I was able to delve and completely understand the ideas behind probabilty and statistics. During Algebra 2 when I first encountered counting principles such as permutations and combinations, I think I was more set on using the equations rather than truly understanding the logic behind the topic which made it more difficult as we began to encounter more complicated/challenging cases and began combining counting principles as in section 12.1.4. I think the text's way of writing out the definition/equation for a permutation was a bit confusing for me to understand at first since I had always approached it a bit differently. I think the ellipses/repeating multiplication symbols within the equation P(n,k) = n(n-1)(n-2) ... (n-k+1) made me forget about having the same number of descending factors (k) as parentheses on the right side of the equation and led me to think that there were more factors in between (n-2) and (n-k+1) that had to be multiplied to calculate the permutation. Although the examples of counting principles have always been centered around marbles in a bag or a deck of cards, I think that encountering it in a life sciences model/scope would be very interesting and more related to the career fields that interest me. I see now that using these counting principles in certain situations could be a much more effective, specific, and logical way of going about solving a real-life problem.

hw 0 syllabus

2006-09-30T03:01:00.000-07:00

Jennifer Han
Math 3c-Lecture 1
Discussion Section D
URL: http://jhanmath3c.blogspot.com

HW 0: I acknowledge that I will be responsible for the content and policies listed in the syllabus for Math 3c for the remainder of the class.

(a) What is your name?
Jennifer Han

(b) What is your discussion section? (You will need this for your homework and the exams.)
Discussion Section D

(c) Even though not a question, give me a random number between 1 and 100.
36

(d) What math classes have you taken at UCLA (given by course number)?
I am a Freshman and have not taken any math classes at UCLA prior to enrolling in Math 3c and Statistics 13A this 2006 Fall Quarter.

(e) What do you like about math, or what is your strength in math?
When I was younger, I used to enjoy math because it felt really rewarding to arrive at the answer. I liked the fact that after tackling a challenging math/word problem and trying to solve it through multiple methods, you could feel a small sense of accomplishment when you discovered your answer was correct. As I take more and more math classes, I am beginning to realize that math is a very broad and complicated subject with many real-life applications and connections, which I think makes the subject much more intriguing.

(f) What do you not like about math, or what is your weakness in math?
I feel that I may be somewhat of a timid student in the beginning which makes me a bit hesitant to ask questions. However, I am hoping that as I become more comfortable in the class, I will be more pro-active in my studying. At times, if I am struggling with a problem and cannot arrive at the correct answer despite multiple attempts and trials, I become intimidated and easily frustrated and begin to doubt if I will ever arrive at the answer. This frustration usually leads me to dislike the problem, the section, etc, but I am hoping that I will learn to be a bit more patient with myself and learn to ask for help when necessary. I have realized that when I am attempting to solve a challenging math problem, my mind seems to make the process much more complicated than it needs to be. I think I become intimidated which makes it more difficult for me to see other possibilities in going about a math problem.

(g) Think of your favorite math teacher: what did (s)he do best? (no names please...)
My favorite math teacher was my geometry teacher during high school, who was very considerate towards his students. I had always enjoyed math before this class, but it was one of the first times that I realized math could be fun due to the multiple projects and hands-on demonstrations in the class. In geometry, we first encountered simple proofs, and as a result of my teacher's help and understanding, I really grew to enjoy solving and writing proofs that year.

(h) Think of your worst math teacher: what did (s)he do worst? (again, no names please...)
My least favorite math teacher spoke in a monotone voice (although I am sure he did not intend to or had no control over his speaking voice) and copied examples and problems from the textbook. Since he showed us the same examples and identical steps as outlined and written in the textbook, many of the students in the class had difficulty figuring out problems that varied from the set examples. I think if he used other examples and references about the same section/topic, the students might have been a bit more engaged or had more resources to learn from. He seemed to recite the information rather than explain the material, which made the class a bit lazy and lose interest in the subject. Everyday, our teacher would ask us if we had any questions about the homework from the night before, and if a student had a question, the teacher would actually solve the entire problem out for the class on the whiteboard. In the end, many of the students took advantage of the teacher and would resort to not completing any of the home work and then coming into class and asking for all the problems that were assigned while copying down the answers word for word at the beginning of the period. Then the students would turn in the assignments that the teacher completed for them while receiving full/partial credit.

(i) What is a semi-late assignment?
A semi-late assignment is a homework assignment that is turned in on the specified due date between 8:10-8:50AM on the day of the lecture. All homework needs to be turned in at the beginning of class approximately between 8:00AM (or earlier) and 8:10AM to be considered on time. The semi-late homework assignment will not be graded for correctness but can still receive a maximum of 10 points out of the 20 points for homework if all the problems are complete (half-credit). However, if a homework assignment is turned in after 8:50AM or after class, it will not be collected and zero points will be given.

(j) When is an assignment complete, and how many points do you lose if the assignment
is not complete?
An assignment is complete if all the assigned problems are finished and worked out in detail. For ever missing problem on a homework assignment, a point will be deducted from the maximum 10 points you can receive for completeness. Another 10 points will be rewarded if the 5 problems (randomly chosen and looked at by the grader) are solved correct.

(k) What is the five minutes rule about?
The 5 minutes rule states that if Prof. Brose is found anywhere at anytime on campus, she will probably have at least 5 minutes to talk to and answer questions for you.

(l) What happens to the final exams after they are graded?
After final exams are graded, they are kept for one quarter, filed for another quarter, and then recycled.