Mudassir Lodi
Homework for Lecture 7 of Dr. Z.’s Dynamical Models in Biology class
Email the answers (either as .pdf file or .txt file) to ShaloshBEkhad@gmail.com by 8:00pm Monday, Sept. 27, 2021. Subject: hw7 with an attachment hw7FirstLast.pdf and/or hw7FirstLast.txt Also please indicate (EITHER way) whether it is OK to post 

1. Use the procedure GRt(p,i,N) in today’s Maple code https://sites.math.rutgers.edu/~zeilberg/Bio21/M7.txt to write a procedure
EstGR(p,i,N,K) that runs GRt(p,i,N)K times, and uses it to estimate (by simulation) the pair [ProbabilityOfExitingAwinner , AverageDurationOfGame]	
EstGR:=proc(p,i,N,K) local X,x,d,c:
X:=RandomVariable(Bernoulli(p)):
c:=0:
x:=i:
while x>0 and x<K do
 c:=c+1:
 d:=trunc(Sample(X,1)[1]):
  if d=0 then
    x:=x-1:
   else
  x:=x+1:
  fi:
 od:
 if x=0 then
   RETURN([0,c]):
  elif x=K then
   RETURN([1,c]):
  fi:
end:

2. (Theoretical question)
a For a fixed N, Let xN(i) be the probability of exiting a winner in a Gambler’s Ruin game with max capital N dollars and entering capital of i dollars, with a fair coin. Explain why the following linear (homog.) recurrence is true
	 	,
is true. Also explain why the following boundary conditions are true
	xN(0) = 0	,	xN(N) = 1	.
The homologous reccurence is true because there are only two outcomes for a fair coin, with equal probability of occurring. Therefore, the coefficient ½ preceding the recurrence equation would be appropriate. The boundary conditions are true for the same reason, as the maximum probability can only be 1 (100%). 
b: Prove that the closed-form expression  satisfies the same recurrence and boundary conditions as xN(i), and explain why we have a beautiful explicit formula xN(i) = Ni .
The inverse would yield the same results, and therefore result in the explicit formula. 
c For a fixed N, Let EN(i) be the expected number of rounds it takes to complete a Gambler’s ruin game (exiting either a winner or loser) Explain why the following linear (inhomog.) recurrence is true
	 	,
is true. Also explain why the following boundary conditions are true
	EN(0) = 0	,	EN(N) = 0	.
The number of rounds it takes to complete the game is independent of the outcome (winner or loser). Therefore, the boundary conditions would not be dependent on a binary outcome. 
d: Prove that the closed-form expression zN(i) = i(N − i) satisfies the same recurrence and boundary conditions as EN(i), and explain why we have a beautiful explicit formula EN(i) = i(N − i)
Substituting i(N-i) would yield the same results, and therefore result in the same explicit formula. 

3.	(Maple question).
Use 2 (even of you were unable to prove things, but I hope that you did) to write a procedure
ExactFairGR(i,N) that outputs the pair
[ProbabilityOfExitingAwinner , AverageDuration] for any i and N (of course 0 ≤ i ≤ N).
See how the outputs of ExactFairGR(i,N) and EstGR(1/2 , i,N,3000) compare to each other, for N = 20 and all i from 1 to 19.

GR:=proc(i,N) local X,x,d,c:
X:=RandomVariable(Bernoulli(p)):
c:=0:
x:=i:
 print(`I enter the casino with `, i, `dollars `):
while x>0 and x<N do
 c:=c+1:
 d:=trunc(Sample(X,1)[1]):
  if d=0 then
    x:=x-1:
   else
  x:=x+1:
  fi:
 od:
 if x=0 then
   RETURN([0,c]):
  elif x=N then
   RETURN([1,c]):
  fi:

4.	Generate 10 random stochastic matrices using RandSM(N) (with N = 10), and compare the outputs of
StSa(P,K), StSp(P,K) (with K = 4000), and the exact StS(P).
StSa:=proc(P,4000) local L,x,f,i:
L:=MarkovTrip(P,4000):
f:=add(x[L[i]],i=1..4000):
evalf([seq(coeff(f,x[i],1)/4000,i=1..nops(P))]):

end:

RandSM:=proc(N) local i:
[seq(RandPV(N),i=1..N)]:

end: