Homework

Homework

• Homework set 1: Due Jan 26
  Problems (Not exercises)
  1.2: 1,6,13
  1.3: 6,7,8a,9,11
  1.4: 3,5
  1.5: 1,7
  Solutions.

• Homework set 2: Due Feb 2
  Problems (Not exercises)
  2.1: 1a,2,4,8
  2.3: 1,3,4 Solutions.

• Homework set 3: Due Feb 9
  Problems (Not exercises)
  3.1: 2,4
  3.2: 1,2,4,5 Solutions.

• Homework set 4: Due Feb 18
  Problem 1. Consider the sequence defined recursively by $a_1=3, a_2=7, a_n=a_{n-1}+a_{n-2},n\geq 3$. Find a closed form formula for $a_n$. Note that if you do this for $a_1=a_2=1$ you get the Fibonacci sequence.
  Problem 2. Let $X_n$ be a Markov Chain with stationary transition probabilities. a) Let $i$ be any state. Prove that $P(X_n=i \text{ for infinitely many indices } n|X_0=j)=0\forall j$ if and only if $P(X_n=i \text{ for infinitely many indices } n|X_0=i)=0$. \ b) Prove that $P(X_n=i \text{ for infinitely many indices } n|X_0=i)=0$ if and only if $P(\exists n>0: X_n=i|X_0=i)\neq 1$.
  Make sure you give rigorous arguments using the Markov Property and note some heuristics.
  Exercise 3.4.8
  Problems
  3.4: 1,2,14,18
  Solutions.

• Homework set 5: Due Feb 25
  Exercises
  3.5.6, 3.6.1
  Problems
  3.4.12,3.4.17,3.5.5 (See page 72 for the definition of a martingale),3.7.1
  Solutions.

• Midterm exam
  Solutions.

• Homework set 6: Due Mar 18
  Problem 1. Let $a_n$ be a sequence such that $\lim_{n\rightarrow\infty}a_n=a$. Prove that $\lim_{m\rightarrow\infty}\frac{a_1+\dots+a_m}{m}=a$ as well.
  Exercises
  3.8.1, 3.8.3(only do n=4)
  Problems
  3.8.2,3.8.3
  Solutions.

• Homework set 7: Due Mar 25
  Exercises
  4.1.1
  Problems
  3.9.1,3.9.8,4.1.1,4.1.4,4.1.6,4.1.13
  Solutions.

• Homework set 8: Due Apr 1
  Exercises
  4.3.1,4.3.4
  Problems
  4.3.2,4.4.4,4.4.5,5.1.1,5.1.12
  Solutions.

• Homework set 9: Due Apr 15
  Problem 1. Let $N$ be a Poisson point process of rate $\lambda$. Let $W_k$ be the time of the $k$’th event. Let $Y_1,Y_2,\dots$ be independent random variables with density $g$. Let $0<t_1<t_2$ and let $h_1<h_2<\dots<h_{k+1}$. Define $N_m$ to be the number of points $(W_i,Y_i)$ in the rectangle $(t_1,t_2]\times(h_m,h_{m+1}]$. Prove that $N_1,\dots,N_k$ are independent and that $N_m$ is a Poisson random variable with rate $\iint_A \lambda g(y)dy dt$.
  Exercises
  5.3.4
  Problems
  5.2.4,5.2.11,5.3.8,5.4.7
  Solutions.

• Homework set 10: Due Apr 27 (NOTE: This is due on Wednesday, as that’s the last day of classes!)
  Exercises
  5.6.1,5.6.4,6.1.3
  Problems
  5.5.4,5.6.1,6.1.1,6.1.8,6.2.1,6.4.2
  Solutions.

• Final exam
  Solutions.