Linear Recursive equations

Question 1

What is a stochastic Process

Answer 1

a collection of random variables indexed by time

Question 2

What is the formula for a linear recursive equation? What does each part mean? What can you do with it?

Answer 2

X(n+2) = a (n+1) + ß (n) : 2nd order general equation
-Used to solve recurrence equations
-can be turned into a matrix of dimension 2

Question 3

What is the general form of a 2nd order recurrence equation when finding the powers? What are we using it for?

Answer 3

Xn = Ca^n
-we use it to solve the eigen values for the matrix
|alpha Beta |
| 1 0 |

-the solutions to this equation are the eigen values
-we also use it to get rid of the recurrence
-now we can love x n=100 directly

Question 4

Background:
What is the background for solving 2nd order recurrence equation?

Answer 4

:

Question 5

How would you solve Xm = Ca^m? How does it relate to eigen values? What is the conditioning of the equation?

Answer 5

1. plug it into the recurrence equation
2. simplify ( we want a polynomial form)
3. Find the discriminant (3 cases)
   +: two real solutions
   -: two complex solutions
   0: one solution (eigen values are the same)
   -a^m and ma^m (where a = alpha/2)
4. we solve for Cs using are solution to the polynomial equation and power equations
5. all solution will be a linear combo of the resulting formula for the two solutions

Question 6

what is the equation for the discriminant?

Answer 6

b^2 - 4ac

Question 7

how do you get from the 2nd order recurrence equation to Xm = Ca^m? hint a proof from 1/24/23

Question 8

How can you check if your solutions forXm = Ca^m are linearly independent?

Question 9

Do 1 Problem

Question 10

How to solve:
X(n+2) = aX(n+1) + B(n) + f(n)

Question 11

What is the difference between:
1. X(n+2) = aX(n+1) + B(n) + f(n)
2. X(n+2) = a (n+1) + ß (n)

What are the differences when solving them?