determining if it is convergent or divergent + taylor series

Bernhard, I have a few questions that i hope you can help me clarify...

1. for finding the normalizing constant, is it integral from a to b of the function = A or integral from a to b of the function TIMES A = 1 and then solve for A?

2. In the problem sets, 7.11 i was wondering why it shouldn't be a permutation since you technically can't choose if you want a head or tail. Q7.11: how many ways are there to get 3 times H and 2 times T by tossing a fair coin 5 times? What is the probability of getting 3 heads in 5 fair coin tosses?

3. for an unfair object, like in APRIL 2011 FINAL EXAM number 2, if it's twice as likely and there's in total 6 possibilities, does that mean you add an extra probability....like hence it results 7 and not 6? because it says the probability for a 6 is 2/7 and for other numbers it's 5/7.

4. What is the use of the normalized binomial in CH7? z= (x-np)/standard deviation

5. is random walker or hardy weinberg genetics going to be on the exam? if so is random walker just the same concept of tossing a coin, just that now it's a person walking? and is there shortcuts to drawing the genotype table so you can solve for the probabilities quicker?

Sorry for all these questions!

Jt21:29, 7 April 2012

Hello Jt, since this is a wiki you don't need to address your questions to me. Anyone is free to answer them.

Let's see:

1. The integral from a to b over the function needs to equal 1. Hence A = 1/integral. See, e.g. here.

2. I'm not sure if I get where your confusion comes from. But a coin toss is the classical example of a Bernoulli distribution. Here p = 0.5, hence q = 1-p = 0.5. If you let Head be a success, then the probability of 3 successes is ${\displaystyle C(5,3)p^{3}q^{2}=C(5,3)(1/2)^{3}(1/2)^{2}=C(5,3)(1/2)^{5}=10/32,}$ as asserted.

3. I'm not sure which question you are referring to. But if the question is something like it is twice as likely that roll a 6 than any other number you do the following: Let a be the probability of any other number. Then 2a is the probability of a 6. The total probability needs to be 1. With this information you can find a: a+a+a+a+a+2a = 1, i.e. a = 1/7 and thus prob(6) = 2/7.

4. The normalized binomial is needed to justify the use of a continuous random variable (with Gaussian as pdf) as an approximation to a discrete random variable (with binomial pdf). So in this sense it motivates chapter 8. In real life this is extremely useful in [Statistical hypothesis testing, but we didn't go into details here so I don't think you should worry about it.

5. Of course I can't comment on your first question. But yes, this is an example that, mathematically, it is the same as tossing a coin. But the application can we very different and easily applicable to a real world problem. Same with Hardy Weinberg genetics, since this relies on the binomial distribution (and is hence mathematically the same as a coin toss) you can apply the theoretical results on the binomial to genetics.

Hope this helps. Your questions are very welcome :)