Homework 4

Choose one of the following. (Please don't do both. Choose the one you that interests you more and do it well.)

• Option 1. Recall that Pascal's Triangle is formed by the binomial coefficients placed in a triangular pattern as follows:
    or    
  In particular, this means that the binomial expansions can be found as follows:
  
  See some curious properties of Pascal's triangle on this page. We have talked about some of them (e.g. symmetry and horizontal sums) in class, but this page contains more. Prove that the third diagonal contains triangular numbers (see the very first section, "Diagonals"). Also prove the "Squares" and "Fibonacci sequence" properties.
• Option 2. Find/write three problems on probability with continuous variables such as the ones in this handout which was shown in class. However, your problems have to be different from those in the handout, not just be slight modifications of those. Solve at least two of them. (If you find or think of an interesting problem that you do not know how to solve, we want to see it too, so that is why you are welcome to have one problem without a solution.)