Solve the following problems:
1. If X, Y are lognormal distribution, what the distribution of X/Y.
2. What is the probability of getting 40 or less heads in 100 coin tosses. What distribution is this. With what other distribution can you approximate this. What is the final probability.
3. Which is greater exp((a+b)/2) or (exp(a)+exp(b))/2
4. What is the series expansion of (1/(1-x))
7. If x and y have correlation .9 and y and z have correlation .9 what correlation can we have between x and z.
9. A rabbit can jump one step or two steps a time, there are N steps, write down a formula to give all the possible ways.
10. There is a fair coin, give the expected value of the first two consecutive heads show up.
11. There are 15 horses, the maximum number of horses in a race is 5, how many races are needed to identify the top 3 fastest horses.
12. There are stacks (first in last out), simulate pipe (first in first out), what's the order.
14. which one is the bigger e^PI or PI^e, no calculators, explain how to estimate.
15. You take a grid with n rows and m columns. You start from the lower left corner and you have to go to the upper right corner. You can't neither go down or left. How many possibilities have you ?
16. We play a game. You have a probability p to earn 1 dollars and a probability (1-p) to lose 1 dollar. You start the game with ten dollars.
What is your probability to lose 3) dollars before loosing everything?
17. Let's play a game. I roll a die three times. After each time, you can decide to stop me and you will earn in dollars the number the die gives. What is your strategy? Can your write a program that will compute your expectation for the same game but n rolls? Describe your thoughts.
18. What is the solution of the following equation? x^x^x^x^x^x and so on to the infinity =2
19. In a box, you have two coins. One has two head sides, the other one has on head side and one tail side. You takes one coins and you throw it. You get head. What is the probability to get a head if you throw it again?