12.6.2 Part 2

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

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

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

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

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

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

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

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

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

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

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.