How Can Probability and Statistics Help Solve These Complex Problems?

[Bunting]

Hiya, this is second year UG Coursework (worth very little) so I don't want any "answers" to the questions (else its plagarism ;)) but hints in the right direction if you will :)

Question 1
A random walker takes one step to the right with probability p times that with which she takes p successive steps to the left. Show that her typical distance from the origin grows like sqrt(p).

Question 2
Variables x1 and x2 are drawn independently from two Gaussian distributions with means µ1 and µ2, and standard deviations sigma1 and sigma2 respectively. What are the mean values and standard deviations of x1 + x2 and x1x2?

I have found the first answer (x1+x2) to this using moment generating functions, but I can't find a way to apply this method to x1x2. edit: I think i have foudn the answer to part 2 as well, in a manner of speaking. It seems some reasearch has been recently written on the subject, which is rather annoying. It didn't help much, but I've got some form of an answer.

Question 3
An archer’s aim has a Gaussian distribution about the centre of a circular target. If the standard distribution is such that he hits the target 50% of the time, by how much must he reduce the standard deviation to hit 90% of the time?

With this can i simply approximate a normal distribution to a binomial distribution ?

Thanks a bunch for any suggestions/pointers in the right direction :)

 

[Greg Bernhardt]

For question 1, you can use the central limit theorem to show that the sum of the steps follows a normal distribution. Then you can calculate the expected value of the sum and from there you can determine the typical distance from the origin.
For question 2, it looks like you have already found the mean values and standard deviations of x1+x2 using moment generating functions. For x1x2, you can use the same approach but you will need to work with the joint distribution of x1 and x2.
For question 3, you can use the normal approximation of the binomial distribution to calculate the required reduction in the standard deviation.