Spring 2018, Math 171

Week 1

I flip a coin with probability \(p\) of heads until the first head occurs. Let \(N\) be the number of times I flip this coin. I then flip a coin with probability \(q\) of heads \(N\) times. Let \(H\) be the number of heads which occur with the second coin.

1. (Discussed) Compute \(P(N=n)\)

2. (Discussed) Compute \(P(H=0)\)

   • Hint: Law of Total Probability

3. (Discussed) Compute \(P(N=n|H=0)\)

   • Hint: Bayes Formula

4. (Discussed) Compute the distribution of \[M(N) = \begin{cases}\frac{N}{2}, &\text{if } n \text{ is even} \\ \frac{N+1}{2}, &\text{if } n \text{ is odd} \end{cases}\]

5. (Discussed) Show memorylessness: \(P(N=m+n|N>m) = P(N=n)\)

   • Hint: Use the definition of Conditional Probability