Problem 2. Consider a random walk on the n cycle {1,2,...,n} that moves anticlockwise with probability 1/2 and clockwis...
1-D Random Walk: Consider a random walk described by the following probability rules: P(+x) = 0.5; P(-x) = 0.1 ; P(ty) = 0.2; P(-y) = 0.2 (a) Is the walk biased? If so in which direction? Explain. (b) Compute the following for N steps if the step size is equal to a: <x, y>, <x>, <y'> (c) After long time (after large number of steps, where would the object be found? (find Ox, Ox I.
1-D Random Walk: Consider a...
1. Random Walk: Consider a random walk described by the following probability rules: P(+x) 0.5; P(-x) 0.1; P(ty) 0.2; P(-y) 0.2 (a) Is the walk biased? If so in which direction? Explain. (b) Compute the following for N steps if the step size is equal to a: <x>, <y>, <x>, <y (c) After long time (after large number of steps, where would the object be found? (find σ, and
1. Random Walk: Consider a random walk described by the following...
n) . ..f 1s a Simple random walk wi Find the probability Ps, = 2, 'Sm.メ3for all m = 1,2, 3 I So = 0).
4. (Dobrow 2.5) Consider a random walk on {0,...,k}, which moves
left and right with respective probabilities q and p. If the walk
is at 0 it transitions to 1 on the next step. If the walk is at k
it transitions to k−1 on the next step. This is called random walk
with reflecting boundaries. Assume that k = 3, q = 1/4, p = 3/4, and
the initial distribution is uniform.
(a) Find the transition matrix.
(b) Find...
(1) Consider the probability space 2 [0, 1. We define the probability of an event A Ω to be its length, we define a sequence random variables as follows: When n is odd Xn (u) 0 otherwise while, when n is even otherwise (a) Compute the PMF and CDF of each Xn (b) Deduce that X converge in distribution (c) Show that for any n and any random variable X : Ω R. (d) Deduce that Xn does not converge...
2. Problem 2.5. Consider a random walk on 10..... which movies left and right with respective probabilities a and p. If the walk is at 0 it transitions to 1 on the next step. If the walk is at k it transitions to k-1 on the next step. This is called random walk with reflecting boundaries. Assume that k 3, =1/4, p = 3/4, and the initial distribution is uniform. For the following, use technology if needed. (a) (10.1.X2 }...
Consider a random walk on {0, 1, . . . , N } with jump
probabilities p(x,x+1)=1/3, p(x,x−1)=2/3 for1≤x≤N−1 p(0,0)=1−c,
p(0,1)=c, p(N,N)=1−c, p(N,N−1)=c with p(x, y) = 0 for all other
cases. (Here c is a fixed number in (0, 1).) Find the expected
return time to 0 if we start the process there.
please answer it asap due in 10 hours. thanks
Consider a random walk on {0,1,...,N} with jump probabilities p(л, т + 1) — 1/3, p(х,т —...
Problem 3 Consider a random walk on the integers. Suppose we start from 0, and at each step, we either go left or right with probability 1/2, ie, Xo--0, and Xt+1 Xt+Zt, where Zt-1 with probability 1/2, and Zt1 with probability 1/2. What is the probability distribution of XT? What is E(X) and Var(XT)?
Problem 3 Consider a random walk on the integers. Suppose we start from 0, and at each step, we either go left or right with probability...
The tanks of a country's army are numbered 1,2,.., N. In a war, this country loses n random tanks to the enemy (where n is a given constant). Let the random variables X1,X2, ..Xn be the numberst of the captured tanks and Y be max{X1, X2,..., Xn). Find the probability mass function of Y and find E(Y) Hint l: (n) is the coefficient of xn in the polynomial (1+p" Hint 2: -n(n) =n(1+x)k is the coefficient of xn in the...
Problem 5.2 (10 points) For the simple symmetric random walk (Sn)n=0.12 that with So = 0, show for all n>0 and all -n<k<n
Problem 5.2 (10 points) For the simple symmetric random walk (Sn)n=0.12 that with So = 0, show for all n>0 and all -n