Alright, here is the Chapter 6 Assignment. Rather than listing the problems in the order they appear, I will list them in order of topic first, and in order of importance within topic.
1. Binomial Distribution Problems
Most Basic
Problems 1, 9, and 5 are the most basic questions on the binomial distribution. I would recommend doing Problems 1 and 9 for sure. Problem 5 is slightly different, so it is probably worth doing as well. (If you haven’t got to the Poisson distribution yet, you can skip that part of Problem 9 for now and come back to it.)
Combined with conditional probability
Problem 6 combines a binomial problem with a conditional probability. It’s worth doing to connect back to Chapter V. It doesn’t introduce anything new, though.
Reversing the logic
Problem 3 reverses the logic: rather giving the length of a sequence and asking for the probability of a given number of “successes”, it gives you the probability you want for a given number of successes and asks how long the sequence needs to be. It’s not central, but it is highly recommended, to help solidify ideas. Note that you will get an inequality for the unknown n, and to solve it, you will need to use logarithms. Please ask if you’re rusty with logarithms and can’t figure it out.
combining with combinatorics
Problems 2 and 4 are similar to problems listed already, but involve more combinatorics. I am not emphasizing combinatorics, so you can safely skip these; but if you like combinatorics, they are interesting to do.
Problem 7 gives a combinatorics problem which can be estimated by a binomial distribution. This is an interesting idea, and would help make your comprehension deeper about the differences between Bernoulli trials and non-Bernoulli trials. However, it’s a bit subtle, and could safely be skipped if you are low on time.
Challenge problems
Problem 19 is an interesting challenge problem, using the binomial distribution. It also leads to an interesting problem about patterns in Pascal’s triangle. I would recommend trying this if you are finding the other binomial problems easy. If you solve it, take a look at the book’s answer: the author is using a pattern in Pascal’s triangle to simplify—see if you can prove the pattern as well!
Problem 18 is a problem that looks easy at first, and then as you get into it is super-challenging! It isn’t the binomial distribution, exactly, but it uses a reasoning similar to how we got the binomial distribution, in a trickier case. It’s a side route, but if you are looking for a challenge I would recommend it!
2. Poisson Distribution
most basic
Problems 10, 11, and 13 are the most straightforward applications of the Poisson distribution. Everyone should do Problem 10; I recommend doing Problem 11 as well, if you have time; Problem 13 is a bit repetitive of 10 and 11, so not essential, but do it if you want more practice.
Reversing the Logic
Problems 12 and 15 “reverse the logic” for the Poisson distribution, in a similar way to Problem 3 for the binomial. I think everyone should do Problem 12. Problem 15 is a repetition of 12, but a little strangely worded; you could skip it unless you want more practice. Again, for Problem 12, you will need to use logarithms to solve for n.
Estimation of Binomial
Problem 9 can also be computed approximately, using the Poisson distribution. You can compare the numbers that you get with the binomial. Since the n is small, the approximation isn’t great, but it is already pretty good. Not essential, but useful to understand better how the Poisson distribution approximates the binomial.
Deceptively simple
I find Problem 14 fascinating and important, and I think everyone should do it. What I find surprising about this problem is that it seems like we don’t have enough information, but in fact we do. It illustrates the power and universality of the Poisson distribution.
Problem 23 is another problem that seems like we don’t have enough information, but we do. Though it uses the Poisson distribution, it doesn’t use the distribution directly, so much as the idea of the expected number of “successes” $\lambda$, and the “law of large numbers”. It’s a bit of a side road, but worth trying if you have time.
testing your understanding
Problem 16 is a good practical question, which combines Poisson and binomial reasoning. It’s a bit challenging: the hard part of this question is figuring out how to set it up. It’s a very good test for understanding, and I recommend it as a “stretch” question for most people (unless you’re still just trying to understand the basics, then you could skip it). Note that you can find a more explicit answer than what the book gives.
Problem 17 is theoretically quite important. I would recommend that everyone at least try to understand what the problem is claiming, because the result of the problem is important. It is a very good test of your understanding to figure out how to set up the problem; if you are feeling confident with the basic concepts, I would recommend trying to get that far. Actually getting to the final answer is an interesting problem in algebra and binomial coefficients, and recommended if you are looking for a fun challenge problem.
For those who know some calculus
It is often important to add up a lot of binomial probabilities for practical questions. For example, if we flip 100 coins, and want to know the probability that the number of heads is less than 40, we have to add up $$\left(\binom{100}{0}+\binom{100}{1}+\binom{100}{2}+\dotsb+\binom{100}{39}\right)\left(\frac{1}{2}\right)^{100}.$$(In this book, this sum is written as $B(39;100,1/2)$; see formula (10.6) on page 173.) This would be untenable by hand, and even for a computer it is computationally pretty intensive. (It gets even worse if I didn’t pick a problem with $p=1/2$, because then I wouldn’t be able to take out a common factor.) In the next chapter we will see a way of estimating sums like this using the normal distribution. That method is usually sufficient, but it doesn’t work well in all cases.
In Problem 45, the author shows how to use calculus to give an exact answer to the previous question, without summing up all those terms. If you know some calculus, I recommend at least trying to understand formula (10.8) (for example, write out what it means in my example above). Note that the formula involves an integral that we cannot do explicitly, so we still need to use a computer to estimate it; but there are very powerful and fast algorithms to estimate integrals, so the problem is much easier for the computer than adding up the series I showed above. Actually proving formula (10.8) is a fun but difficult calculus problem—try it if you want to see a very non-trivial application of calculus!
Problem 46 does the same thing as Problem 45, except for the Poisson distribution, rather than the binomial distribution. I have the same recommendations: if you know some calculus, I highly recommend at least trying to understand what formula (10.10) is telling you, and to think of a problem where it could be applied. If you are looking for a calculus challenge, try to prove (10.10)!
