Stochastic process consisting both coding and proof, and some computation for constructing a transition matrix

MATH 5040 – HW 3

Due back at 9 pm on Saturday, 31st October.

-1- Recall the example of the chess board Knight that travels along a random path on 3 X 3 grid. Write the transition matrix for the path of the Knight, find π̄ and E{T1

ii }.

-2- Consider the random path for a chess board Rook on a 2 X 2 grid. Write the transition matrix for the path of the Knight, find π̄ and E{T1

ii }.

-3- Show that the invariant probability π(j) = (π̄P)j as given in the video for example of Urn Model 1 (under the Canvas page for Week 7 of the course).

-4- Find the transition probabilities and construct P for example of Urn Model 2 in video under the Canvas page for Week 7 of the course.

-5- Refer to the virus mutation problem introduced in class on 8th October. Show that:

Pn(0, 0) = 1

N +

[ 1 −

Nα

N − 1

]n [ 1 −

1

N

] You could simplify calculations in the recursive formula by letting a = α

N−1 and b = 1 − αN N−1.

-6- Consider the characteristic polynomial of a linear difference equation (from class on 8th October): bu2 −u + a = 0. We know that the roots of this equation are given by:

u1,2 = 1 ± √

1 − 4ab 2b

Suppose 1 − 4ab 6= 0 and u1 = x + iy and u2 = ū1. Show that the solution of form: f(n) = α1u

n 1 + α2u

n 2 can be written in polar form as: fn = r

n[c1 cos(nθ) + c2 sin(nθ)].

-7- Suppose we want to generate the following discrete random variables from U ∼ Unif(0, 1). Draw a graph and indicate G(u) values for 4 different choices of u values in each case. (from class on 13th October)

-a- X ∼ Ber(p = 0.25). -b- X ∼ Bin(n = 2,p = 0.25). -c- X ∼ Discrete Uniform{1, 2, . . . , 6}.

-8- Write a program in each of the above cases to generate 100 samples from each distribution. Plot a histogram and compute the sample mean and variance in each case. Provide files with code separately. (from class on 13th October)

-9- Derive the expression to generate samples from Exponential(λ = 2) using a random number gener- ator that generates U ∼Unif(0, 1). Write a program to generate 100 samples from Exponential(λ = 2) (from class on 13th October).

-10- Consider a simple symmetric random walk in Z2. Derive an expression for ∞∑ n=0

P2n(0, 0) (from class

on 22nd October).