Homework Answers
E[S12]=1*0.5=0.5
VAR[S12]=E[S12^2]-E[S12]^2=1^2*0.5-0.5^2=0.25
Mn=e^(2Sn)
M7= e^(2S7)
E[M7]=e^(2S7)*p(S7)=e^-2*0.3=0.3*e^-2
V[M7]=E[M7^2]-E[M7]^2=e^-4*0.3-0.09*e^-4=0.21*e^-4
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1. For a random walk So, S1,. . . starting from So=-1 with i.i.d. increments which...
1. Random Walk: Consider a random walk described by the following probability rules: P(+x) 0.5; P...
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...
1-D Random Walk: Consider a random walk described by the following probability rules: P(+x) = 0.5...
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...
A discrete random variable A takes values {1, 2, 4} with probabilities specified as follows: P[A...
A discrete random variable A takes values {1, 2, 4} with probabilities specified as follows: P[A = 1] = 0.5, P[A = 2] = 0.3 and P [A = 4] = 0.2 Given A= ), a discrete random variable N is Poisson distributed with rate equal to 1, that is: 9 P[N = n|A = 1] = in n! el Hint If N is Poisson distributed with rate 1, its expectation and variance are as follows: E[N] = Var [N]...
Code in Python Problem 1 (2 Points) 1. Write a function randomWalk(.. .) which simulates one path (or trajectory) of a s...
Code in Python
Problem 1 (2 Points) 1. Write a function randomWalk(.. .) which simulates one path (or trajectory) of a simple symmetric random walk with 2N time steps (i.e. from 0,1,2,...,2N) starting at So-0 nput: lengthofRandomWalk2N Output: samplePath: Array of length 2N+1 with the entire path of the random walk on 0,1,2,..,2N In def randomwalk(lengthofRandomwalk): ## WRITE YOUR OWN CODE HERE # HINT: USE np. random . choice ( ) TO SIMULATE THE INCREMENTS return samplePath In [ ]:...
7. A positive random variable Y is said to be a lognormal random variable, LOGN (u,...
7. A positive random variable Y is said to be a lognormal random variable, LOGN (u, 0), if In Y ~ N(No?). We assume that Y, LOGN (1,0%), i = 1,..., n are independent. [5] (a) Find the distribution of T = 11",Y. [4] (b) Find E(T) and Var(T) (5] (c) If we assume that M = ... = Hn and a = ... = 0, what does the the successive geometric average, lim (II",Y), converge in probability to? Justify...
Code in Python Problem 1 (2 Points) 1. Write a function randomWalk(.. .) which simulates one...
Code in Python
Problem 1 (2 Points) 1. Write a function randomWalk(.. .) which simulates one path (or trajectory) of a simple symmetric random walk with 2N time steps (i.e. from 0,1,2,...,2N) starting at So-0 nput: lengthofRandomWalk2N Output: samplePath: Array of length 2N+1 with the entire path of the random walk on 0,1,2,..,2N In def randomwalk(lengthofRandomwalk): ## WRITE YOUR OWN CODE HERE # HINT: USE np. random . choice ( ) TO SIMULATE THE INCREMENTS return samplePath In [ ]:...
N (,02). We 7. A positive random variable Y is said to be a lognormal random...
N (,02). We 7. A positive random variable Y is said to be a lognormal random variable, LOGN(1,02), if In Y assume that Y, LOGN(Mi, 0), i = 1,...,n are independent. [5] (a) Find the distribution of T = II Y. [4] (b) Find E(T) and Var(T) 5) (c) If we assume that Hi = ... = Hn and oi = ... = on what does the the successive geometric average, lim (IIYA), converge in probability to? Justify your answer....
4. (Dobrow 2.5) Consider a random walk on {0,...,k}, which moves left and right with respective...
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...
in 4. Suppose that {Xk, k > 1} is a sequence of i.i.d. random variables with...
in 4. Suppose that {Xk, k > 1} is a sequence of i.i.d. random variables with P(X1 = +1) = 1. Let Sn = 2h=1 Xk (i.e. Sn, n > 1 is a symmetric simple random walk with steps Xk, k > 1). (a) Compute E[S+1|X1, ... , Xn] for n > 1. Hint: Check out Example 3.8 in the lecture notes (Version Mar/04/2020) for inspiration. (b) Find deterministic coefficients an, bn, Cn possibly depending on n so that Mn...
3. Assume that the lifetimes (measured from the beginning of use) of light bulbs are i.i.d....
3. Assume that the lifetimes (measured from the beginning of use) of light bulbs are i.i.d. random variables with distribution P(T> k) = (k +1)-, k = 0,1,2, ..., for some B > 0. (Note that time is measured in discrete units.) In a lightbulb socket in a factory, a bulb is used until it fails, and then it is replaced at the next time unit. Let (Xn)n>o be the irreducible Markov chain which records the age of the bulb...
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...
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...
A discrete random variable A takes values {1, 2, 4} with probabilities specified as follows: P[A = 1] = 0.5, P[A = 2] = 0.3 and P [A = 4] = 0.2 Given A= ), a discrete random variable N is Poisson distributed with rate equal to 1, that is: 9 P[N = n|A = 1] = in n! el Hint If N is Poisson distributed with rate 1, its expectation and variance are as follows: E[N] = Var [N]...
Code in Python
Problem 1 (2 Points) 1. Write a function randomWalk(.. .) which simulates one path (or trajectory) of a simple symmetric random walk with 2N time steps (i.e. from 0,1,2,...,2N) starting at So-0 nput: lengthofRandomWalk2N Output: samplePath: Array of length 2N+1 with the entire path of the random walk on 0,1,2,..,2N In def randomwalk(lengthofRandomwalk): ## WRITE YOUR OWN CODE HERE # HINT: USE np. random . choice ( ) TO SIMULATE THE INCREMENTS return samplePath In [ ]:...
7. A positive random variable Y is said to be a lognormal random variable, LOGN (u, 0), if In Y ~ N(No?). We assume that Y, LOGN (1,0%), i = 1,..., n are independent. [5] (a) Find the distribution of T = 11",Y. [4] (b) Find E(T) and Var(T) (5] (c) If we assume that M = ... = Hn and a = ... = 0, what does the the successive geometric average, lim (II",Y), converge in probability to? Justify...
Code in Python
Problem 1 (2 Points) 1. Write a function randomWalk(.. .) which simulates one path (or trajectory) of a simple symmetric random walk with 2N time steps (i.e. from 0,1,2,...,2N) starting at So-0 nput: lengthofRandomWalk2N Output: samplePath: Array of length 2N+1 with the entire path of the random walk on 0,1,2,..,2N In def randomwalk(lengthofRandomwalk): ## WRITE YOUR OWN CODE HERE # HINT: USE np. random . choice ( ) TO SIMULATE THE INCREMENTS return samplePath In [ ]:...
N (,02). We 7. A positive random variable Y is said to be a lognormal random variable, LOGN(1,02), if In Y assume that Y, LOGN(Mi, 0), i = 1,...,n are independent. [5] (a) Find the distribution of T = II Y. [4] (b) Find E(T) and Var(T) 5) (c) If we assume that Hi = ... = Hn and oi = ... = on what does the the successive geometric average, lim (IIYA), converge in probability to? Justify your answer....
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...
in 4. Suppose that {Xk, k > 1} is a sequence of i.i.d. random variables with P(X1 = +1) = 1. Let Sn = 2h=1 Xk (i.e. Sn, n > 1 is a symmetric simple random walk with steps Xk, k > 1). (a) Compute E[S+1|X1, ... , Xn] for n > 1. Hint: Check out Example 3.8 in the lecture notes (Version Mar/04/2020) for inspiration. (b) Find deterministic coefficients an, bn, Cn possibly depending on n so that Mn...
3. Assume that the lifetimes (measured from the beginning of use) of light bulbs are i.i.d. random variables with distribution P(T> k) = (k +1)-, k = 0,1,2, ..., for some B > 0. (Note that time is measured in discrete units.) In a lightbulb socket in a factory, a bulb is used until it fails, and then it is replaced at the next time unit. Let (Xn)n>o be the irreducible Markov chain which records the age of the bulb...
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